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### 6 Manipulating Codes

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)).

#### 6.1 Functions that Generate a New Code from a Given Code

##### 6.1-1 ExtendedCode
 ‣ 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.

##### 6.1-2 PuncturedCode
 ‣ 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.

##### 6.1-3 EvenWeightSubcode
 ‣ 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)).

##### 6.1-4 PermutedCode
 ‣ 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


##### 6.1-5 ExpurgatedCode
 ‣ 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)).

##### 6.1-6 AugmentedCode
 ‣ 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)).

##### 6.1-7 RemovedElementsCode
 ‣ 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)).

##### 6.1-9 ShortenedCode
 ‣ 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.

##### 6.1-10 LengthenedCode
 ‣ 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.

##### 6.1-11 SubCode
 ‣ 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 ]


##### 6.1-12 ResidueCode
 ‣ 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


##### 6.1-13 ConstructionBCode
 ‣ 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


##### 6.1-14 DualCode
 ‣ 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


##### 6.1-15 ConversionFieldCode
 ‣ 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 ] ]


##### 6.1-16 TraceCode
 ‣ 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



##### 6.1-17 CosetCode
 ‣ 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


##### 6.1-18 ConstantWeightSubcode
 ‣ 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 ]


##### 6.1-19 StandardFormCode
 ‣ 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 ] ]


##### 6.1-20 PiecewiseConstantCode
 ‣ 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.

#### 6.2 Functions that Generate a New Code from Two or More Given Codes

##### 6.2-1 DirectSumCode
 ‣ 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


##### 6.2-2 UUVCode
 ‣ 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


##### 6.2-3 DirectProductCode
 ‣ 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


##### 6.2-4 IntersectionCode
 ‣ 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, C) = C;
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)).

##### 6.2-5 UnionCode
 ‣ 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


##### 6.2-6 ExtendedDirectSumCode
 ‣ 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


##### 6.2-7 AmalgamatedDirectSumCode
 ‣ 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> f;
a linear [22,8,3]7 amalgamated direct sum code


##### 6.2-8 BlockwiseDirectSumCode
 ‣ 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


##### 6.2-9 ConstructionXCode
 ‣ 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 $$d = \min\{ D_j, D_{j-1} + d_{j-1}, D_{j-2} + d_{j-2} + d_{j-1}, \ldots, D_1 + \sum_{i=1}^{j-1} d_i\}$$.

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) ]);;
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, C);
true
gap> IsSubset(C, C);
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 ]" ]


##### 6.2-10 ConstructionXXCode
 ‣ 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

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" ]


##### 6.2-11 BZCode
 ‣ 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.

##### 6.2-12 BZCodeNC
 ‣ 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, Cy, Cy ];
[ 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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