6 Manipulating Codes

6.1
Functions that Generate a New Code from a Given Code

6.1-1 ExtendedCode

6.1-2 PuncturedCode

6.1-3 EvenWeightSubcode

6.1-4 PermutedCode

6.1-5 ExpurgatedCode

6.1-6 AugmentedCode

6.1-7 RemovedElementsCode

6.1-8 AddedElementsCode

6.1-9 ShortenedCode

6.1-10 LengthenedCode

6.1-11 SubCode

6.1-12 ResidueCode

6.1-13 ConstructionBCode

6.1-14 DualCode

6.1-15 ConversionFieldCode

6.1-16 TraceCode

6.1-17 CosetCode

6.1-18 ConstantWeightSubcode

6.1-19 StandardFormCode

6.1-20 PiecewiseConstantCode

6.1-1 ExtendedCode

6.1-2 PuncturedCode

6.1-3 EvenWeightSubcode

6.1-4 PermutedCode

6.1-5 ExpurgatedCode

6.1-6 AugmentedCode

6.1-7 RemovedElementsCode

6.1-8 AddedElementsCode

6.1-9 ShortenedCode

6.1-10 LengthenedCode

6.1-11 SubCode

6.1-12 ResidueCode

6.1-13 ConstructionBCode

6.1-14 DualCode

6.1-15 ConversionFieldCode

6.1-16 TraceCode

6.1-17 CosetCode

6.1-18 ConstantWeightSubcode

6.1-19 StandardFormCode

6.1-20 PiecewiseConstantCode

In this chapter we describe several functions **GUAVA** uses to manipulate codes. Some of the best codes are obtained by starting with for example a BCH code, and manipulating it.

In some cases, it is faster to perform calculations with a manipulated code than to use the original code. For example, if the dimension of the code is larger than half the word length, it is generally faster to compute the weight distribution by first calculating the weight distribution of the dual code than by directly calculating the weight distribution of the original code. The size of the dual code is smaller in these cases.

Because **GUAVA** keeps all information in a code record, in some cases the information can be preserved after manipulations. Therefore, computations do not always have to start from scratch.

In Section 6.1, we describe functions that take a code with certain parameters, modify it in some way and return a different code (see `ExtendedCode`

(6.1-1), `PuncturedCode`

(6.1-2), `EvenWeightSubcode`

(6.1-3), `PermutedCode`

(6.1-4), `ExpurgatedCode`

(6.1-5), `AugmentedCode`

(6.1-6), `RemovedElementsCode`

(6.1-7), `AddedElementsCode`

(6.1-8), `ShortenedCode`

(6.1-9), `LengthenedCode`

(6.1-10), `ResidueCode`

(6.1-12), `ConstructionBCode`

(6.1-13), `DualCode`

(6.1-14), `ConversionFieldCode`

(6.1-15), `ConstantWeightSubcode`

(6.1-18), `StandardFormCode`

(6.1-19) and `CosetCode`

(6.1-17)). In Section 6.2, we describe functions that generate a new code out of two codes (see `DirectSumCode`

(6.2-1), `UUVCode`

(6.2-2), `DirectProductCode`

(6.2-3), `IntersectionCode`

(6.2-4) and `UnionCode`

(6.2-5)).

`‣ ExtendedCode` ( C[, i] ) | ( function ) |

`ExtendedCode`

extends the code `C` `i` times and returns the result. `i` is equal to \(1\) by default. Extending is done by adding a parity check bit after the last coordinate. The coordinates of all codewords now add up to zero. In the binary case, each codeword has even weight.

The word length increases by `i`. The size of the code remains the same. In the binary case, the minimum distance increases by one if it was odd. In other cases, that is not always true.

A cyclic code in general is no longer cyclic after extending.

gap> C1 := HammingCode( 3, GF(2) ); a linear [7,4,3]1 Hamming (3,2) code over GF(2) gap> C2 := ExtendedCode( C1 ); a linear [8,4,4]2 extended code gap> IsEquivalent( C2, ReedMullerCode( 1, 3 ) ); true gap> List( AsSSortedList( C2 ), WeightCodeword ); [ 0, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8 ] gap> C3 := EvenWeightSubcode( C1 ); a linear [7,3,4]2..3 even weight subcode

To undo extending, call `PuncturedCode`

(see `PuncturedCode`

(6.1-2)). The function `EvenWeightSubcode`

(see `EvenWeightSubcode`

(6.1-3)) also returns a related code with only even weights, but without changing its word length.

`‣ PuncturedCode` ( C ) | ( function ) |

`PuncturedCode`

punctures `C` in the last column, and returns the result. Puncturing is done simply by cutting off the last column from each codeword. This means the word length decreases by one. The minimum distance in general also decrease by one.

This command can also be called with the syntax `PuncturedCode( C, L )`

. In this case, `PuncturedCode`

punctures `C` in the columns specified by `L`, a list of integers. All columns specified by `L` are omitted from each codeword. If \(l\) is the length of `L` (so the number of removed columns), the word length decreases by \(l\). The minimum distance can also decrease by \(l\) or less.

Puncturing a cyclic code in general results in a non-cyclic code. If the code is punctured in all the columns where a word of minimal weight is unequal to zero, the dimension of the resulting code decreases.

gap> C1 := BCHCode( 15, 5, GF(2) ); a cyclic [15,7,5]3..5 BCH code, delta=5, b=1 over GF(2) gap> C2 := PuncturedCode( C1 ); a linear [14,7,4]3..5 punctured code gap> ExtendedCode( C2 ) = C1; false gap> PuncturedCode( C1, [1,2,3,4,5,6,7] ); a linear [8,7,1]1 punctured code gap> PuncturedCode( WholeSpaceCode( 4, GF(5) ) ); a linear [3,3,1]0 punctured code # The dimension decreased from 4 to 3

`ExtendedCode`

extends the code again (see `ExtendedCode`

(6.1-1)), although in general this does not result in the old code.

`‣ EvenWeightSubcode` ( C ) | ( function ) |

`EvenWeightSubcode`

returns the even weight subcode of `C`, consisting of all codewords of `C` with even weight. If `C` is a linear code and contains words of odd weight, the resulting code has a dimension of one less. The minimum distance always increases with one if it was odd. If `C` is a binary cyclic code, and \(g(x)\) is its generator polynomial, the even weight subcode either has generator polynomial \(g(x)\) (if \(g(x)\) is divisible by \(x-1\)) or \(g(x)\cdot (x-1)\) (if no factor \(x-1\) was present in \(g(x)\)). So the even weight subcode is again cyclic.

Of course, if all codewords of `C` are already of even weight, the returned code is equal to `C`.

gap> C1 := EvenWeightSubcode( BCHCode( 8, 4, GF(3) ) ); an (8,33,4..8)3..8 even weight subcode gap> List( AsSSortedList( C1 ), WeightCodeword ); [ 0, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 6, 4, 4, 6, 4, 4, 8, 6, 4, 6, 8, 4, 4, 4, 6, 4, 6, 8, 4, 6, 8 ] gap> EvenWeightSubcode( ReedMullerCode( 1, 3 ) ); a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2)

`ExtendedCode`

also returns a related code of only even weights, but without reducing its dimension (see `ExtendedCode`

(6.1-1)).

`‣ PermutedCode` ( C, L ) | ( function ) |

`PermutedCode`

returns `C` after column permutations. `L` (in GAP disjoint cycle notation) is the permutation to be executed on the columns of `C`. If `C` is cyclic, the result in general is no longer cyclic. If a permutation results in the same code as `C`, this permutation belongs to the automorphism group of `C` (see `AutomorphismGroup`

(4.4-3)). In any case, the returned code is equivalent to `C` (see `IsEquivalent`

(4.4-1)).

gap> C1 := PuncturedCode( ReedMullerCode( 1, 4 ) ); a linear [15,5,7]5 punctured code gap> C2 := BCHCode( 15, 7, GF(2) ); a cyclic [15,5,7]5 BCH code, delta=7, b=1 over GF(2) gap> C2 = C1; false gap> p := CodeIsomorphism( C1, C2 ); ( 2, 4,14, 9,13, 7,11,10, 6, 8,12, 5) gap> C3 := PermutedCode( C1, p ); a linear [15,5,7]5 permuted code gap> C2 = C3; true

`‣ ExpurgatedCode` ( C, L ) | ( function ) |

`ExpurgatedCode`

expurgates the code `C`> by throwing away codewords in list `L`. `C` must be a linear code. `L` must be a list of codeword input. The generator matrix of the new code no longer is a basis for the codewords specified by `L`. Since the returned code is still linear, it is very likely that, besides the words of `L`, more codewords of `C` are no longer in the new code.

gap> C1 := HammingCode( 4 );; WeightDistribution( C1 ); [ 1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1 ] gap> L := Filtered( AsSSortedList(C1), i -> WeightCodeword(i) = 3 );; gap> C2 := ExpurgatedCode( C1, L ); a linear [15,4,3..4]5..11 code, expurgated with 7 word(s) gap> WeightDistribution( C2 ); [ 1, 0, 0, 0, 14, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]

This function does not work on non-linear codes. For removing words from a non-linear code, use `RemovedElementsCode`

(see `RemovedElementsCode`

(6.1-7)). For expurgating a code of all words of odd weight, use `EvenWeightSubcode' (see `EvenWeightSubcode`

(6.1-3)).

`‣ AugmentedCode` ( C, L ) | ( function ) |

`AugmentedCode`

returns `C` after augmenting. `C` must be a linear code, `L` must be a list of codeword inputs. The generator matrix of the new code is a basis for the codewords specified by `L` as well as the words that were already in code `C`. Note that the new code in general will consist of more words than only the codewords of `C` and the words `L`. The returned code is also a linear code.

This command can also be called with the syntax `AugmentedCode(C)`

. When called without a list of codewords, `AugmentedCode`

returns `C` after adding the all-ones vector to the generator matrix. `C` must be a linear code. If the all-ones vector was already in the code, nothing happens and a copy of the argument is returned. If `C` is a binary code which does not contain the all-ones vector, the complement of all codewords is added.

gap> C31 := ReedMullerCode( 1, 3 ); a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2) gap> C32 := AugmentedCode(C31,["00000011","00000101","00010001"]); a linear [8,7,1..2]1 code, augmented with 3 word(s) gap> C32 = ReedMullerCode( 2, 3 ); true gap> C1 := CordaroWagnerCode(6); a linear [6,2,4]2..3 Cordaro-Wagner code over GF(2) gap> Codeword( [0,0,1,1,1,1] ) in C1; true gap> C2 := AugmentedCode( C1 ); a linear [6,3,1..2]2..3 code, augmented with 1 word(s) gap> Codeword( [1,1,0,0,0,0] ) in C2; true

The function `AddedElementsCode`

adds elements to the codewords instead of adding them to the basis (see `AddedElementsCode`

(6.1-8)).

`‣ RemovedElementsCode` ( C, L ) | ( function ) |

`RemovedElementsCode`

returns code `C` after removing a list of codewords `L` from its elements. `L` must be a list of codeword input. The result is an unrestricted code.

gap> C1 := HammingCode( 4 );; WeightDistribution( C1 ); [ 1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1 ] gap> L := Filtered( AsSSortedList(C1), i -> WeightCodeword(i) = 3 );; gap> C2 := RemovedElementsCode( C1, L ); a (15,2013,3..15)2..15 code with 35 word(s) removed gap> WeightDistribution( C2 ); [ 1, 0, 0, 0, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1 ] gap> MinimumDistance( C2 ); 3 # C2 is not linear, so the minimum weight does not have to # be equal to the minimum distance

Adding elements to a code is done by the function `AddedElementsCode`

(see `AddedElementsCode`

(6.1-8)). To remove codewords from the base of a linear code, use `ExpurgatedCode`

(see `ExpurgatedCode`

(6.1-5)).

`‣ AddedElementsCode` ( C, L ) | ( function ) |

`AddedElementsCode`

returns code `C` after adding a list of codewords `L` to its elements. `L` must be a list of codeword input. The result is an unrestricted code.

gap> C1 := NullCode( 6, GF(2) ); a cyclic [6,0,6]6 nullcode over GF(2) gap> C2 := AddedElementsCode( C1, [ "111111" ] ); a (6,2,1..6)3 code with 1 word(s) added gap> IsCyclicCode( C2 ); true gap> C3 := AddedElementsCode( C2, [ "101010", "010101" ] ); a (6,4,1..6)2 code with 2 word(s) added gap> IsCyclicCode( C3 ); true

To remove elements from a code, use `RemovedElementsCode`

(see `RemovedElementsCode`

(6.1-7)). To add elements to the base of a linear code, use `AugmentedCode`

(see `AugmentedCode`

(6.1-6)).

`‣ ShortenedCode` ( C[, L] ) | ( function ) |

`ShortenedCode( C )`

returns the code `C` shortened by taking a cross section. If `C` is a linear code, this is done by removing all codewords that start with a non-zero entry, after which the first column is cut off. If `C` was a \([n,k,d]\) code, the shortened code generally is a \([n-1,k-1,d]\) code. It is possible that the dimension remains the same; it is also possible that the minimum distance increases.

If `C` is a non-linear code, `ShortenedCode`

first checks which finite field element occurs most often in the first column of the codewords. The codewords not starting with this element are removed from the code, after which the first column is cut off. The resulting shortened code has at least the same minimum distance as `C`.

This command can also be called using the syntax `ShortenedCode(C,L)`

. When called in this format, `ShortenedCode`

repeats the shortening process on each of the columns specified by `L`. `L` therefore is a list of integers. The column numbers in `L` are the numbers as they are before the shortening process. If `L` has \(l\) entries, the returned code has a word length of \(l\) positions shorter than `C`.

gap> C1 := HammingCode( 4 ); a linear [15,11,3]1 Hamming (4,2) code over GF(2) gap> C2 := ShortenedCode( C1 ); a linear [14,10,3]2 shortened code gap> C3 := ElementsCode( ["1000", "1101", "0011" ], GF(2) ); a (4,3,1..4)2 user defined unrestricted code over GF(2) gap> MinimumDistance( C3 ); 2 gap> C4 := ShortenedCode( C3 ); a (3,2,2..3)1..2 shortened code gap> AsSSortedList( C4 ); [ [ 0 0 0 ], [ 1 0 1 ] ] gap> C5 := HammingCode( 5, GF(2) ); a linear [31,26,3]1 Hamming (5,2) code over GF(2) gap> C6 := ShortenedCode( C5, [ 1, 2, 3 ] ); a linear [28,23,3]2 shortened code gap> OptimalityLinearCode( C6 ); 0

The function `LengthenedCode`

lengthens the code again (only for linear codes), see `LengthenedCode`

(6.1-10). In general, this is not exactly the inverse function.

`‣ LengthenedCode` ( C[, i] ) | ( function ) |

`LengthenedCode( C )`

returns the code `C` lengthened. `C` must be a linear code. First, the all-ones vector is added to the generator matrix (see `AugmentedCode`

(6.1-6)). If the all-ones vector was already a codeword, nothing happens to the code. Then, the code is extended `i` times (see `ExtendedCode`

(6.1-1)). `i` is equal to \(1\) by default. If `C` was an \([n,k]\) code, the new code generally is a \([n+i,k+1]\) code.

gap> C1 := CordaroWagnerCode( 5 ); a linear [5,2,3]2 Cordaro-Wagner code over GF(2) gap> C2 := LengthenedCode( C1 ); a linear [6,3,2]2..3 code, lengthened with 1 column(s)

`ShortenedCode`

' shortens the code, see `ShortenedCode`

(6.1-9). In general, this is not exactly the inverse function.

`‣ SubCode` ( C[, s] ) | ( function ) |

This function `SubCode`

returns a subcode of `C` by taking the first \(k - s\) rows of the generator matrix of `C`, where \(k\) is the dimension of `C`. The interger `s` may be omitted and in this case it is assumed as 1.

gap> C := BCHCode(31,11); a cyclic [31,11,11]7..11 BCH code, delta=11, b=1 over GF(2) gap> S1:= SubCode(C); a linear [31,10,11]7..13 subcode gap> WeightDistribution(S1); [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 120, 190, 0, 0, 272, 255, 0, 0, 120, 66, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> S2:= SubCode(C, 8); a linear [31,3,11]14..20 subcode gap> History(S2); [ "a linear [31,3,11]14..20 subcode of", "a cyclic [31,11,11]7..11 BCH code, delta=11, b=1 over GF(2)" ] gap> WeightDistribution(S2); [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

`‣ ResidueCode` ( C[, c] ) | ( function ) |

The function `ResidueCode`

takes a codeword `c` of `C` (if `c` is omitted, a codeword of minimal weight is used). It removes this word and all its linear combinations from the code and then punctures the code in the coordinates where `c` is unequal to zero. The resulting code is an \([n-w, k-1, d-\lfloor w*(q-1)/q \rfloor ]\) code. `C` must be a linear code and `c` must be non-zero. If `c` is not in ` then no change is made to ``C`.

gap> C1 := BCHCode( 15, 7 ); a cyclic [15,5,7]5 BCH code, delta=7, b=1 over GF(2) gap> C2 := ResidueCode( C1 ); a linear [8,4,4]2 residue code gap> c := Codeword( [ 0,0,0,1,0,0,1,1,0,1,0,1,1,1,1 ], C1);; gap> C3 := ResidueCode( C1, c ); a linear [7,4,3]1 residue code

`‣ ConstructionBCode` ( C ) | ( function ) |

The function `ConstructionBCode`

takes a binary linear code `C` and calculates the minimum distance of the dual of `C` (see `DualCode`

(6.1-14)). It then removes the columns of the parity check matrix of `C` where a codeword of the dual code of minimal weight has coordinates unequal to zero. The resulting matrix is a parity check matrix for an \([n-dd, k-dd+1, \geq d]\) code, where \(dd\) is the minimum distance of the dual of `C`.

gap> C1 := ReedMullerCode( 2, 5 ); a linear [32,16,8]6 Reed-Muller (2,5) code over GF(2) gap> C2 := ConstructionBCode( C1 ); a linear [24,9,8]5..10 Construction B (8 coordinates) gap> BoundsMinimumDistance( 24, 9, GF(2) ); rec( n := 24, k := 9, q := 2, references := rec( ), construction := [ [ Operation "UUVCode" ], [ [ [ Operation "UUVCode" ], [ [ [ Operation "DualCode" ], [ [ [ Operation "RepetitionCode" ], [ 6, 2 ] ] ] ], [ [ Operation "CordaroWagnerCode" ], [ 6 ] ] ] ], [ [ Operation "CordaroWagnerCode" ], [ 12 ] ] ] ], lowerBound := 8, lowerBoundExplanation := [ "Lb(24,9)=8, u u+v construction of C1 and C2:", "Lb(12,7)=4, u u+v construction of C1 and C2:", "Lb(6,5)=2, dual of the repetition code", "Lb(6,2)=4, Cordaro-Wagner code", "Lb(12,2)=8, Cordaro-Wagner code" ], upperBound := 8, upperBoundExplanation := [ "Ub(24,9)=8, otherwise construction B would contradict:", "Ub(18,4)=8, Griesmer bound" ] ) # so C2 is optimal

`‣ DualCode` ( C ) | ( function ) |

`DualCode`

returns the dual code of `C`. The dual code consists of all codewords that are orthogonal to the codewords of `C`. If `C` is a linear code with generator matrix \(G\), the dual code has parity check matrix \(G\) (or if `C` has parity check matrix \(H\), the dual code has generator matrix \(H\)). So if `C` is a linear \([n, k]\) code, the dual code of `C` is a linear \([n, n-k]\) code. If `C` is a cyclic code with generator polynomial \(g(x)\), the dual code has the reciprocal polynomial of \(g(x)\) as check polynomial.

The dual code is always a linear code, even if `C` is non-linear.

If a code `C` is equal to its dual code, it is called *self-dual*.

gap> R := ReedMullerCode( 1, 3 ); a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2) gap> RD := DualCode( R ); a linear [8,4,4]2 Reed-Muller (1,3) code over GF(2) gap> R = RD; true gap> N := WholeSpaceCode( 7, GF(4) ); a cyclic [7,7,1]0 whole space code over GF(4) gap> DualCode( N ) = NullCode( 7, GF(4) ); true

`‣ ConversionFieldCode` ( C ) | ( function ) |

`ConversionFieldCode`

returns the code obtained from `C` after converting its field. If the field of `C` is \(GF(q^m)\), the returned code has field \(GF(q)\). Each symbol of every codeword is replaced by a concatenation of \(m\) symbols from \(GF(q)\). If `C` is an \((n, M, d_1)\) code, the returned code is a \((n\cdot m, M, d_2)\) code, where \(d_2 > d_1\).

See also `HorizontalConversionFieldMat`

(7.3-10).

gap> R := RepetitionCode( 4, GF(4) ); a cyclic [4,1,4]3 repetition code over GF(4) gap> R2 := ConversionFieldCode( R ); a linear [8,2,4]3..4 code, converted to basefield GF(2) gap> Size( R ) = Size( R2 ); true gap> GeneratorMat( R ); [ [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ] ] gap> GeneratorMat( R2 ); [ [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ] ]

`‣ TraceCode` ( C ) | ( function ) |

Input: `C` is a linear code defined over an extension \(E\) of `F` (`F` is the ``base field'')

Output: The linear code generated by \(Tr_{E/F}(c)\), for all \(c \in C\).

`TraceCode`

returns the image of the code `C` under the trace map. If the field of `C` is \(GF(q^m)\), the returned code has field \(GF(q)\).

Very slow. It does not seem to be easy to related the parameters of the trace code to the original except in the ``Galois closed'' case.

gap> C:=RandomLinearCode(10,4,GF(4)); MinimumDistance(C); a [10,4,?] randomly generated code over GF(4) 5 gap> trC:=TraceCode(C,GF(2)); MinimumDistance(trC); a linear [10,7,1]1..3 user defined unrestricted code over GF(2) 1

`‣ CosetCode` ( C, w ) | ( function ) |

`CosetCode`

returns the coset of a code `C` with respect to word `w`. `w` must be of the codeword type. Then, `w` is added to each codeword of `C`, yielding the elements of the new code. If `C` is linear and `w` is an element of `C`, the new code is equal to `C`, otherwise the new code is an unrestricted code.

Generating a coset is also possible by simply adding the word `w` to `C`. See 4.2.

gap> H := HammingCode(3, GF(2)); a linear [7,4,3]1 Hamming (3,2) code over GF(2) gap> c := Codeword("1011011");; c in H; false gap> C := CosetCode(H, c); a (7,16,3)1 coset code gap> List(AsSSortedList(C), el-> Syndrome(H, el)); [ [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ], [ 1 1 1 ] ] # All elements of the coset have the same syndrome in H

`‣ ConstantWeightSubcode` ( C, w ) | ( function ) |

`ConstantWeightSubcode`

returns the subcode of `C` that only has codewords of weight `w`. The resulting code is a non-linear code, because it does not contain the all-zero vector.

This command also can be called with the syntax `ConstantWeightSubcode(C)`

In this format, `ConstantWeightSubcode`

returns the subcode of `C` consisting of all minimum weight codewords of `C`.

`ConstantWeightSubcode`

first checks if Leon's binary `wtdist`

exists on your computer (in the default directory). If it does, then this program is called. Otherwise, the constant weight subcode is computed using a GAP program which checks each codeword in `C` to see if it is of the desired weight.

gap> N := NordstromRobinsonCode();; WeightDistribution(N); [ 1, 0, 0, 0, 0, 0, 112, 0, 30, 0, 112, 0, 0, 0, 0, 0, 1 ] gap> C := ConstantWeightSubcode(N, 8); a (16,30,6..16)5..8 code with codewords of weight 8 gap> WeightDistribution(C); [ 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> eg := ExtendedTernaryGolayCode();; WeightDistribution(eg); [ 1, 0, 0, 0, 0, 0, 264, 0, 0, 440, 0, 0, 24 ] gap> C := ConstantWeightSubcode(eg); a (12,264,6..12)3..6 code with codewords of weight 6 gap> WeightDistribution(C); [ 0, 0, 0, 0, 0, 0, 264, 0, 0, 0, 0, 0, 0 ]

`‣ StandardFormCode` ( C ) | ( function ) |

`StandardFormCode`

returns `C` after putting it in standard form. If `C` is a non-linear code, this means the elements are organized using lexicographical order. This means they form a legal GAP `Set'.

If `C` is a linear code, the generator matrix and parity check matrix are put in standard form. The generator matrix then has an identity matrix in its left part, the parity check matrix has an identity matrix in its right part. Although **GUAVA** always puts both matrices in a standard form using `BaseMat`

, this never alters the code. `StandardFormCode`

even applies column permutations if unavoidable, and thereby changes the code. The column permutations are recorded in the construction history of the new code (see `Display`

(4.6-3)). `C` and the new code are of course equivalent.

If `C` is a cyclic code, its generator matrix cannot be put in the usual upper triangular form, because then it would be inconsistent with the generator polynomial. The reason is that generating the elements from the generator matrix would result in a different order than generating the elements from the generator polynomial. This is an unwanted effect, and therefore `StandardFormCode`

just returns a copy of `C` for cyclic codes.

gap> G := GeneratorMatCode( Z(2) * [ [0,1,1,0], [0,1,0,1], [0,0,1,1] ], "random form code", GF(2) ); a linear [4,2,1..2]1..2 random form code over GF(2) gap> Codeword( GeneratorMat( G ) ); [ [ 0 1 0 1 ], [ 0 0 1 1 ] ] gap> Codeword( GeneratorMat( StandardFormCode( G ) ) ); [ [ 1 0 0 1 ], [ 0 1 0 1 ] ]

`‣ PiecewiseConstantCode` ( part, wts[, F] ) | ( function ) |

`PiecewiseConstantCode`

returns a code with length \(n = \sum n_i\), where `part`=\([ n_1, \dots, n_k ]\). `wts` is a list of `constraints` \(w=(w_1,...,w_k)\), each of length \(k\), where \(0 \leq w_i \leq n_i\). The default field is \(GF(2)\).

A constraint is a list of integers, and a word \(c = ( c_1, \dots, c_k )\) (according to `part`, i.e., each \(c_i\) is a subword of length \(n_i\)) is in the resulting code if and only if, for some constraint \(w \in\) `wts`, \(\|c_i\| = w_i\) for all \(1 \leq i \leq k\), where \(\| ...\|\) denotes the Hamming weight.

An example might make things clearer:

gap> PiecewiseConstantCode( [ 2, 3 ], [ [ 0, 0 ], [ 0, 3 ], [ 1, 0 ], [ 2, 2 ] ],GF(2) ); the C code programs are compiled, so using Leon's binary.... the C code programs are compiled, so using Leon's binary.... the C code programs are compiled, so using Leon's binary.... the C code programs are compiled, so using Leon's binary.... a (5,7,1..5)1..5 piecewise constant code over GF(2) gap> AsSSortedList(last); [ [ 0 0 0 0 0 ], [ 0 0 1 1 1 ], [ 0 1 0 0 0 ], [ 1 0 0 0 0 ], [ 1 1 0 1 1 ], [ 1 1 1 0 1 ], [ 1 1 1 1 0 ] ]

The first constraint is satisfied by codeword 1, the second by codeword 2, the third by codewords 3 and 4, and the fourth by codewords 5, 6 and 7.

`‣ DirectSumCode` ( C1, C2 ) | ( function ) |

`DirectSumCode`

returns the direct sum of codes `C1` and `C2`. The direct sum code consists of every codeword of `C1` concatenated by every codeword of `C2`. Therefore, if `Ci` was a \((n_i,M_i,d_i)\) code, the result is a \((n_1+n_2,M_1*M_2,min(d_1,d_2))\) code.

If both `C1` and `C2` are linear codes, the result is also a linear code. If one of them is non-linear, the direct sum is non-linear too. In general, a direct sum code is not cyclic.

Performing a direct sum can also be done by adding two codes (see Section 4.2). Another often used method is the `u, u+v'-construction, described in `UUVCode`

(6.2-2).

gap> C1 := ElementsCode( [ [1,0], [4,5] ], GF(7) );; gap> C2 := ElementsCode( [ [0,0,0], [3,3,3] ], GF(7) );; gap> D := DirectSumCode(C1, C2);; gap> AsSSortedList(D); [ [ 1 0 0 0 0 ], [ 1 0 3 3 3 ], [ 4 5 0 0 0 ], [ 4 5 3 3 3 ] ] gap> D = C1 + C2; # addition = direct sum true

`‣ UUVCode` ( C1, C2 ) | ( function ) |

`UUVCode`

returns the so-called \((u\|u+v)\) construction applied to `C1` and `C2`. The resulting code consists of every codeword \(u\) of `C1` concatenated by the sum of \(u\) and every codeword \(v\) of `C2`. If `C1` and `C2` have different word lengths, sufficient zeros are added to the shorter code to make this sum possible. If `Ci` is a \((n_i,M_i,d_i)\) code, the result is an \((n_1+max(n_1,n_2),M_1\cdot M_2,min(2\cdot d_1,d_2))\) code.

If both `C1` and `C2` are linear codes, the result is also a linear code. If one of them is non-linear, the UUV sum is non-linear too. In general, a UUV sum code is not cyclic.

The function `DirectSumCode`

returns another sum of codes (see `DirectSumCode`

(6.2-1)).

gap> C1 := EvenWeightSubcode(WholeSpaceCode(4, GF(2))); a cyclic [4,3,2]1 even weight subcode gap> C2 := RepetitionCode(4, GF(2)); a cyclic [4,1,4]2 repetition code over GF(2) gap> R := UUVCode(C1, C2); a linear [8,4,4]2 U U+V construction code gap> R = ReedMullerCode(1,3); true

`‣ DirectProductCode` ( C1, C2 ) | ( function ) |

`DirectProductCode`

returns the direct product of codes `C1` and `C2`. Both must be linear codes. Suppose `Ci` has generator matrix \(G_i\). The direct product of `C1` and `C2` then has the Kronecker product of \(G_1\) and \(G_2\) as the generator matrix (see the GAP command `KroneckerProduct`

).

If `Ci` is a \([n_i, k_i, d_i]\) code, the direct product then is an \([n_1\cdot n_2,k_1\cdot k_2,d_1\cdot d_2]\) code.

gap> L1 := LexiCode(10, 4, GF(2)); a linear [10,5,4]2..4 lexicode over GF(2) gap> L2 := LexiCode(8, 3, GF(2)); a linear [8,4,3]2..3 lexicode over GF(2) gap> D := DirectProductCode(L1, L2); a linear [80,20,12]20..45 direct product code

`‣ IntersectionCode` ( C1, C2 ) | ( function ) |

`IntersectionCode`

returns the intersection of codes `C1` and `C2`. This code consists of all codewords that are both in `C1` and `C2`. If both codes are linear, the result is also linear. If both are cyclic, the result is also cyclic.

gap> C := CyclicCodes(7, GF(2)); [ a cyclic [7,7,1]0 enumerated code over GF(2), a cyclic [7,6,1..2]1 enumerated code over GF(2), a cyclic [7,3,1..4]2..3 enumerated code over GF(2), a cyclic [7,0,7]7 enumerated code over GF(2), a cyclic [7,3,1..4]2..3 enumerated code over GF(2), a cyclic [7,4,1..3]1 enumerated code over GF(2), a cyclic [7,1,7]3 enumerated code over GF(2), a cyclic [7,4,1..3]1 enumerated code over GF(2) ] gap> IntersectionCode(C[6], C[8]) = C[7]; true

The *hull* of a linear code is the intersection of the code with its dual code. In other words, the hull of \(C\) is `IntersectionCode(C, DualCode(C))`

.

`‣ UnionCode` ( C1, C2 ) | ( function ) |

`UnionCode`

returns the union of codes `C1` and `C2`. This code consists of the union of all codewords of `C1` and `C2` and all linear combinations. Therefore this function works only for linear codes. The function `AddedElementsCode`

can be used for non-linear codes, or if the resulting code should not include linear combinations. See `AddedElementsCode`

(6.1-8). If both arguments are cyclic, the result is also cyclic.

gap> G := GeneratorMatCode([[1,0,1],[0,1,1]]*Z(2)^0, GF(2)); a linear [3,2,1..2]1 code defined by generator matrix over GF(2) gap> H := GeneratorMatCode([[1,1,1]]*Z(2)^0, GF(2)); a linear [3,1,3]1 code defined by generator matrix over GF(2) gap> U := UnionCode(G, H); a linear [3,3,1]0 union code gap> c := Codeword("010");; c in G; false gap> c in H; false gap> c in U; true

`‣ ExtendedDirectSumCode` ( L, B, m ) | ( function ) |

The extended direct sum construction is described in section V of Graham and Sloane [GS85]. The resulting code consists of `m` copies of `L`, extended by repeating the codewords of `B` `m` times.

Suppose `L` is an \([n_L, k_L]r_L\) code, and `B` is an \([n_L, k_B]r_B\) code (non-linear codes are also permitted). The length of `B` must be equal to the length of `L`. The length of the new code is \(n = m n_L\), the dimension (in the case of linear codes) is \(k \leq m k_L + k_B\), and the covering radius is \(r \leq \lfloor m \Psi( L, B ) \rfloor\), with

\[ \Psi( L, B ) = \max_{u \in F_2^{n_L}} \frac{1}{2^{k_B}} \sum_{v \in B} {\rm d}( L, v + u ). \]

However, this computation will not be executed, because it may be too time consuming for large codes.

If \(L \subseteq B\), and \(L\) and \(B\) are linear codes, the last copy of `L` is omitted. In this case the dimension is \(k = m k_L + (k_B - k_L)\).

gap> c := HammingCode( 3, GF(2) ); a linear [7,4,3]1 Hamming (3,2) code over GF(2) gap> d := WholeSpaceCode( 7, GF(2) ); a cyclic [7,7,1]0 whole space code over GF(2) gap> e := ExtendedDirectSumCode( c, d, 3 ); a linear [21,15,1..3]2 3-fold extended direct sum code

`‣ AmalgamatedDirectSumCode` ( c1, c2[, check] ) | ( function ) |

`AmalgamatedDirectSumCode`

returns the amalgamated direct sum of the codes `c1` and `c2`. The amalgamated direct sum code consists of all codewords of the form \((u \, \| \,0 \, \| \, v)\) if \((u \, \| \, 0) \in c_1\) and \((0 \, \| \, v) \in c_2\) and all codewords of the form \((u \, \| \, 1 \, \| \, v)\) if \((u \, \| \, 1) \in c_1\) and \((1 \, \| \, v) \in c_2\). The result is a code with length \( n = n_1 + n_2 - 1 \) and size \( M \leq M_1 \cdot M_2 / 2 \).

If both codes are linear, they will first be standardized, with information symbols in the last and first coordinates of the first and second code, respectively.

If `c1` is a normal code (see `IsNormalCode`

(7.4-5)) with the last coordinate acceptable (see `IsCoordinateAcceptable`

(7.4-3)), and `c2` is a normal code with the first coordinate acceptable, then the covering radius of the new code is \(r \leq r_1 + r_2 \). However, checking whether a code is normal or not is a lot of work, and almost all codes seem to be normal. Therefore, an option `check` can be supplied. If `check` is true, then the codes will be checked for normality. If `check` is false or omitted, then the codes will not be checked. In this case it is assumed that they are normal. Acceptability of the last and first coordinate of the first and second code, respectively, is in the last case also assumed to be done by the user.

gap> c := HammingCode( 3, GF(2) ); a linear [7,4,3]1 Hamming (3,2) code over GF(2) gap> d := ReedMullerCode( 1, 4 ); a linear [16,5,8]6 Reed-Muller (1,4) code over GF(2) gap> e := DirectSumCode( c, d ); a linear [23,9,3]7 direct sum code gap> f := AmalgamatedDirectSumCode( c, d );; gap> MinimumDistance( f );; gap> CoveringRadius( f );; gap> f; a linear [22,8,3]7 amalgamated direct sum code

`‣ BlockwiseDirectSumCode` ( C1, L1, C2, L2 ) | ( function ) |

`BlockwiseDirectSumCode`

returns a subcode of the direct sum of `C1` and `C2`. The fields of `C1` and `C2` must be same. The lists `L1` and `L2` are two equally long with elements from the ambient vector spaces of `C1` and `C2`, respectively, *or* `L1` and `L2` are two equally long lists containing codes. The union of the codes in `L1` and `L2` must be `C1` and `C2`, respectively.

In the first case, the blockwise direct sum code is defined as

\[ bds = \bigcup_{1 \leq i \leq \ell} ( C_1 + (L_1)_i ) \oplus ( C_2 + (L_2)_i ), \]

where \(\ell\) is the length of `L1` and `L2`, and \(\oplus\) is the direct sum.

In the second case, it is defined as

\[ bds = \bigcup_{1 \leq i \leq \ell} ( (L_1)_i \oplus (L_2)_i ). \]

The length of the new code is \(n = n_1 + n_2\).

gap> C1 := HammingCode( 3, GF(2) );; gap> C2 := EvenWeightSubcode( WholeSpaceCode( 6, GF(2) ) );; gap> BlockwiseDirectSumCode( C1, [[ 0,0,0,0,0,0,0 ],[ 1,0,1,0,1,0,0 ]], > C2, [[ 0,0,0,0,0,0 ],[ 1,0,1,0,1,0 ]] ); a (13,1024,1..13)1..2 blockwise direct sum code

`‣ ConstructionXCode` ( C, A ) | ( function ) |

Consider a list of \(j\) linear codes of the same length \(N\) over the same field \(F\), \(C = \{ C_1, C_2, \ldots, C_j \}\), where the parameter of the \(i\)th code is \(C_i = [N, K_i, D_i]\) and \(C_j \subset C_{j-1} \subset \ldots \subset C_2 \subset C_1\). Consider a list of \(j-1\) auxiliary linear codes of the same field \(F\), \(A = \{ A_1, A_2, \ldots, A_{j-1} \}\) where the parameter of the \(i\)th code \(A_i\) is \([n_i, k_i=(K_i-K_{i+1}), d_i]\), an \([n, K_1, d]\) linear code over field \(F\) can be constructed where \(n = N + \sum_{i=1}^{j-1} n_i\), and \(d = \min\{ D_j, D_{j-1} + d_{j-1}, D_{j-2} + d_{j-2} + d_{j-1}, \ldots, D_1 + \sum_{i=1}^{j-1} d_i\}\).

For more information on Construction X, refer to [SRC72].

gap> C1 := BCHCode(127, 43); a cyclic [127,29,43]31..59 BCH code, delta=43, b=1 over GF(2) gap> C2 := BCHCode(127, 47); a cyclic [127,22,47..51]36..63 BCH code, delta=47, b=1 over GF(2) gap> C3 := BCHCode(127, 55); a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF(2) gap> G1 := ShallowCopy( GeneratorMat(C2) );; gap> Append(G1, [ GeneratorMat(C1)[23] ]);; gap> C1 := GeneratorMatCode(G1, GF(2)); a linear [127,23,1..43]35..63 code defined by generator matrix over GF(2) gap> MinimumDistance(C1); 43 gap> C := [ C1, C2, C3 ]; [ a linear [127,23,43]35..63 code defined by generator matrix over GF(2), a cyclic [127,22,47..51]36..63 BCH code, delta=47, b=1 over GF(2), a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF(2) ] gap> IsSubset(C[1], C[2]); true gap> IsSubset(C[2], C[3]); true gap> A := [ RepetitionCode(4, GF(2)), EvenWeightSubcode( QRCode(17, GF(2)) ) ]; [ a cyclic [4,1,4]2 repetition code over GF(2), a cyclic [17,8,6]3..6 even weight subcode ] gap> CX := ConstructionXCode(C, A); a linear [148,23,53]43..74 Construction X code gap> History(CX); [ "a linear [148,23,53]43..74 Construction X code of", "Base codes: [ a cyclic [127,15,55]41..62 BCH code, delta=55, b=1 over GF(2)\ , a cyclic [127,22,47..51]36..63 BCH code, delta=47, b=1 over GF(2), a linear \ [127,23,43]35..63 code defined by generator matrix over GF(2) ]", "Auxiliary codes: [ a cyclic [4,1,4]2 repetition code over GF(2), a cyclic [\ 17,8,6]3..6 even weight subcode ]" ]

`‣ ConstructionXXCode` ( C1, C2, C3, A1, A2 ) | ( function ) |

Consider a set of linear codes over field \(F\) of the same length, \(n\), \(C_1=[n, k_1, d_1]\), \(C_2=[n, k_2, d_2]\) and \(C_3=[n, k_3, d_3]\) such that \(C_2 \subset C_1\), \(C_3 \subset C_1\) and \(C_4 = C_2 \cap C_3\). Given two auxiliary codes \(A_1=[n_1, k_1-k_2, e_1]\) and \(A_2=[n_2, k_1-k_3, e_2]\) over the same field \(F\), there exists an \([n+n_1+n_2, k_1, d]\) linear code \(C_{XX}\) over field \(F\), where \(d = \min\{d_4, d_3 + e_1, d_2 + e_2, d_1 + e_1 + e_2\}\).

The codewords of \(C_{XX}\) can be partitioned into three sections \(( v\;\|\;a\;\|\;b )\) where \(v\) has length \(n\), \(a\) has length \(n_1\) and \(b\) has length \(n_2\). A codeword from Construction XX takes the following form:

\(( v \; \| \; 0 \; \| \; 0 )\) if \(v \in C_4\)

\(( v \; \| \; a_1 \; \| \; 0 )\) if \(v \in C_3 \backslash C_4\)

\(( v \; \| \; 0 \; \| \; a_2 )\) if \(v \in C_2 \backslash C_4\)

\(( v \; \| \; a_1 \; \| \; a_2 )\) otherwise

For more information on Construction XX, refer to [All84].

gap> a := PrimitiveRoot(GF(32)); Z(2^5) gap> f0 := MinimalPolynomial( GF(2), a^0 ); x_1+Z(2)^0 gap> f1 := MinimalPolynomial( GF(2), a^1 ); x_1^5+x_1^2+Z(2)^0 gap> f5 := MinimalPolynomial( GF(2), a^5 ); x_1^5+x_1^4+x_1^2+x_1+Z(2)^0 gap> C2 := CheckPolCode( f0 * f1, 31, GF(2) );; MinimumDistance(C2);; Display(C2); a cyclic [31,6,15]10..13 code defined by check polynomial over GF(2) gap> C3 := CheckPolCode( f0 * f5, 31, GF(2) );; MinimumDistance(C3);; Display(C3); a cyclic [31,6,15]10..13 code defined by check polynomial over GF(2) gap> C1 := UnionCode(C2, C3);; MinimumDistance(C1);; Display(C1); a linear [31,11,11]7..11 union code of U: a cyclic [31,6,15]10..13 code defined by check polynomial over GF(2) V: a cyclic [31,6,15]10..13 code defined by check polynomial over GF(2) gap> A1 := BestKnownLinearCode( 10, 5, GF(2) ); a linear [10,5,4]2..4 shortened code gap> A2 := DualCode( RepetitionCode(6, GF(2)) ); a cyclic [6,5,2]1 dual code gap> CXX:= ConstructionXXCode(C1, C2, C3, A1, A2 ); a linear [47,11,15..17]13..23 Construction XX code gap> MinimumDistance(CXX); 17 gap> History(CXX); [ "a linear [47,11,17]13..23 Construction XX code of", "C1: a cyclic [31,11,11]7..11 union code", "C2: a cyclic [31,6,15]10..13 code defined by check polynomial over GF(2)", "C3: a cyclic [31,6,15]10..13 code defined by check polynomial over GF(2)", "A1: a linear [10,5,4]2..4 shortened code", "A2: a cyclic [6,5,2]1 dual code" ]

`‣ BZCode` ( O, I ) | ( function ) |

Given a set of outer codes of the same length \(O_i = [N, K_i, D_i]\) over GF(\(q^{e_i}\)), where \(i=1,2,\ldots,t\) and a set of inner codes of the same length \(I_i = [n, k_i, d_i]\) over GF(\(q\)), `BZCode`

returns a Blokh-Zyablov multilevel concatenated code with parameter \([ n \times N, \sum_{i=1}^t e_i \times K_i, \min_{i=1,\ldots,t}\{d_i \times D_i\} ]\) over GF(\(q\)).

Note that the set of inner codes must satisfy chain condition, i.e. \(I_1 = [n, k_1, d_1] \subset I_2=[n, k_2, d_2] \subset \ldots \subset I_t=[n, k_t, d_t]\) where \(0=k_0 < k_1 < k_2 < \ldots < k_t\). The dimension of the inner codes must satisfy the condition \(e_i = k_i - k_{i-1}\), where GF(\(q^{e_i}\)) is the field of the \(i\)th outer code.

For more information on Blokh-Zyablov multilevel concatenated code, refer to [Bro98].

`‣ BZCodeNC` ( O, I ) | ( function ) |

This function is the same as `BZCode`

, except this version is faster as it does not estimate the covering radius of the code. Users are encouraged to use this version unless you are working on very small codes.

gap> # gap> # Binary code gap> # gap> O := [ CyclicMDSCode(2,3,7), BestKnownLinearCode(9,5,GF(2)), CyclicMDSCode(2,3,4) ]; [ a cyclic [9,7,3]1 MDS code over GF(8), a linear [9,5,3]2..3 shortened code, a cyclic [9,4,6]4..5 MDS code over GF(8) ] gap> A := ExtendedCode( HammingCode(3,GF(2)) );; gap> I := [ SubCode(A), A, DualCode( RepetitionCode(8, GF(2)) ) ]; [ a linear [8,3,4]3..4 subcode, a linear [8,4,4]2 extended code, a cyclic [8,7,2]1 dual code ] gap> C := BZCodeNC(O, I); a linear [72,38,12]0..72 Blokh Zyablov concatenated code gap> # gap> # Non binary code gap> # gap> O2 := ExtendedCode(GoppaCode(ConwayPolynomial(5,2), Elements(GF(5))));; gap> O3 := ExtendedCode(GoppaCode(ConwayPolynomial(5,3), Elements(GF(5))));; gap> O1 := DualCode( O3 );; gap> MinimumDistance(O1);; MinimumDistance(O2);; MinimumDistance(O3);; gap> Cy := CyclicCodes(5, GF(5));; gap> for i in [4, 5] do; MinimumDistance(Cy[i]);; od; gap> O := [ O1, O2, O3 ]; [ a linear [6,4,3]1 dual code, a linear [6,3,4]2..3 extended code, a linear [6,2,5]3..4 extended code ] gap> I := [ Cy[5], Cy[4], Cy[3] ]; [ a cyclic [5,1,5]3..4 enumerated code over GF(5), a cyclic [5,2,4]2..3 enumerated code over GF(5), a cyclic [5,3,1..3]2 enumerated code over GF(5) ] gap> C := BZCodeNC( O, I ); a linear [30,9,5..15]0..30 Blokh Zyablov concatenated code gap> MinimumDistance(C); 15 gap> History(C); [ "a linear [30,9,15]0..30 Blokh Zyablov concatenated code of", "Inner codes: [ a cyclic [5,1,5]3..4 enumerated code over GF(5), a cyclic [5\ ,2,4]2..3 enumerated code over GF(5), a cyclic [5,3,1..3]2 enumerated code ove\ r GF(5) ]", "Outer codes: [ a linear [6,4,3]1 dual code, a linear [6,3,4]2..3 extended c\ ode, a linear [6,2,5]3..4 extended code ]" ]

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