# Online manual of codessystem.lib

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## 1. codessystem_lib

Library:

codessystem.lib

Purpose:

Procedures for the lecture 'Codes und Systemtheorie'

Authors:

Daniel Andres, daniel.andres@math.rwth-aachen.de

Guide:

The procedures in this library are implemented in a straight forward way. Therefore, this library is not meant for serious computation, but rather to provide a tool to check some ideas.

References:

Script for the lecture 'Codes und Systemtheorie' by G. Nebe and E. Zerz, RWTH Aachen University, Summer term 2011

Main procedures:

Auxiliary procedures:

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#### 1.1 hammingWeightEnumerator

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

hammingWeightEnumerator(C,p[,v]); C matrix, p int, v optional intvec

Return:

poly, the Hamming weight enumerator of C

Purpose:

compute the Hamming weight enumerator of a linear code

Assume:

C is the generator matrix of a linear code over the prime field of characteristic p>0.

Note:

The optional parameter v can be used to specify the two variables of the basering, which are used to create the Hamming weight enumerator. By default, the first two variables are used.

Display:

If printlevel>0, progress status messages will be printed.

Example:

 ```LIB "codessystem.lib"; ring r = 0,(x,y,z),dp; matrix C[2][4] = 1,1,1,0, 0,1,2,1; hammingWeightEnumerator(C,3); → x4+8xy3 intvec v = 1,2; hammingWeightEnumerator(C,4,v); → // Second argument should be a prime number. → // Using 3 instead. → x4+8xy3 v = 2,3; hammingWeightEnumerator(C,3,v); → y4+8yz3 ```

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#### 1.2 completeWeightEnumerator

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

completeWeightEnumerator(C,p); C matrix, p int

Return:

poly, the complete weight enumerator of C

Purpose:

compute the complete weight enumerator of a linear code

Assume:

C is the generator matrix of a linear code over the prime field of characteristic p>0.
The basering must have at least p variables.

Note:

The first variable of the basering corresponds to the variable belonging to 0 in the ground field, the second variable to 1, .. the p-th variable to p-1.

Display:

If printlevel>0, progress status messages will be printed.

Example:

 ```LIB "codessystem.lib"; ring r = 0,(x0,x1,x2),dp; matrix C[2][4] = 1,1,1,0, 0,1,2,1; completeWeightEnumerator(C,3); → x0^4+x0*x1^3+3*x0*x1^2*x2+3*x0*x1*x2^2+x0*x2^3 ```

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#### 1.3 hammingWeight

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

hammingWeight(c); c matrix

Return:

int, the Hamming Weight of c

Purpose:

compute the Hamming Weight, i.e. the number of non-zero entries

Assume:

the matrix c has only one row

Example:

 ```LIB "codessystem.lib"; ring r = 3,x,dp; matrix c[1][4] = 1,2,3; print(c); → 1,-1,0,0 hammingWeight(c); → 2 ```

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#### 1.4 dualCode

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

dualCode(C); C matrix

Return:

matrix, generator matrix of dual code

Purpose:

compute the dual code

Assume:

the ground field is a prime field of positive characteristic

Example:

 ```LIB "codessystem.lib"; ring r = 3,x,dp; matrix C[2][4] = 1,1,1,0, 0,1,2,1; matrix D = dualCode(C); print(D); → 1, 1,1,0, → -1,0,1,1 ```

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#### 1.5 isSelfDual

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

isSelfDual(C); C matrix

Return:

int, 1 if C generates a self-dual code, 0 otherwise

Purpose:

check whether a linear code is self-dual

Assume:

the ground field is a prime field of positive characteristic

Example:

 ```LIB "codessystem.lib"; ring r = 3,x,dp; matrix C[2][4] = 1,1,1,0, 0,1,2,1; isSelfDual(C); → 1 matrix C2[1][4] = 1,1,1,0; isSelfDual(C2); → 0 ```

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#### 1.6 completeCode

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

completeCode(C); C matrix

Return:

list of matrices

Purpose:

compute all elements of linear space spanned by rows of C

Assume:

the ground field is a prime field of positive characteristic

Example:

 ```LIB "codessystem.lib"; ring r = 3,x,dp; matrix C[2][4] = 1,1,1,0, 0,1,2,1; completeCode(C); → [1]: → _[1,1]=1 → _[1,2]=1 → _[1,3]=1 → _[1,4]=0 → [2]: → _[1,1]=-1 → _[1,2]=-1 → _[1,3]=-1 → _[1,4]=0 → [3]: → _[1,1]=1 → _[1,2]=-1 → _[1,3]=0 → _[1,4]=1 → [4]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → [5]: → _[1,1]=-1 → _[1,2]=0 → _[1,3]=1 → _[1,4]=1 → [6]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=-1 → _[1,4]=-1 → [7]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=-1 → _[1,4]=1 → [8]: → _[1,1]=-1 → _[1,2]=1 → _[1,3]=0 → _[1,4]=-1 → [9]: → _[1,1]=0 → _[1,2]=-1 → _[1,3]=1 → _[1,4]=-1 ```

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#### 1.7 simpleOrbits

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

simpleOrbits(m,G,gm); m matrix, G list of matrices, gm string

Return:

list, the orbit of m under G

Assume:

- elements in G generate a (finite) matrix group,
- the string gm defines a valid group action, e.g. 'g*m' or
'inverse(g)*m*g', where the group element is always denoted by g
and the set member always by m

Purpose:

compute the orbit of m under the action gm of the group G

Display:

If printlevel>0, progress status messages will be printed.

Example:

 ```LIB "codessystem.lib"; ring r = 0,x,dp; matrix A[4][4]= 0,0,0,1, 1,0,0,0, 0,1,0,0, 0,0,1,0; matrix B[4][4]= 0,1,0,0, 1,0,0,0, 0,0,1,0, 0,0,0,1; list G = A,B; // G generates S_4 simpleOrbits(A,G,"m*g"); // all elements of S_4 → [1]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=1 → _[2,1]=1 → _[2,2]=0 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=1 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=1 → _[4,4]=0 → [2]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=1 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=0 → _[2,4]=1 → _[3,1]=1 → _[3,2]=0 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=1 → _[4,3]=0 → _[4,4]=0 → [3]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=1 → _[2,1]=0 → _[2,2]=1 → _[2,3]=0 → _[2,4]=0 → _[3,1]=1 → _[3,2]=0 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=1 → _[4,4]=0 → [4]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=1 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=0 → _[3,4]=1 → _[4,1]=1 → _[4,2]=0 → _[4,3]=0 → _[4,4]=0 → [5]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=1 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=0 → _[2,4]=1 → _[3,1]=0 → _[3,2]=1 → _[3,3]=0 → _[3,4]=0 → _[4,1]=1 → _[4,2]=0 → _[4,3]=0 → _[4,4]=0 → [6]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=1 → _[1,4]=0 → _[2,1]=1 → _[2,2]=0 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=0 → _[3,4]=1 → _[4,1]=0 → _[4,2]=1 → _[4,3]=0 → _[4,4]=0 → [7]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=1 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=1 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=0 → _[4,4]=1 → [8]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=1 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=0 → _[3,4]=1 → _[4,1]=0 → _[4,2]=1 → _[4,3]=0 → _[4,4]=0 → [9]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=1 → _[2,4]=0 → _[3,1]=1 → _[3,2]=0 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=0 → _[4,4]=1 → [10]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=0 → _[2,4]=1 → _[3,1]=0 → _[3,2]=0 → _[3,3]=1 → _[3,4]=0 → _[4,1]=1 → _[4,2]=0 → _[4,3]=0 → _[4,4]=0 → [11]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=1 → _[1,4]=0 → _[2,1]=0 → _[2,2]=1 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=0 → _[3,4]=1 → _[4,1]=1 → _[4,2]=0 → _[4,3]=0 → _[4,4]=0 → [12]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=0 → _[1,4]=0 → _[2,1]=1 → _[2,2]=0 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=1 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=0 → _[4,4]=1 → [13]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=1 → _[2,1]=0 → _[2,2]=1 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=1 → _[3,4]=0 → _[4,1]=1 → _[4,2]=0 → _[4,3]=0 → _[4,4]=0 → [14]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=1 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=0 → _[3,4]=1 → _[4,1]=0 → _[4,2]=0 → _[4,3]=1 → _[4,4]=0 → [15]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=1 → _[2,4]=0 → _[3,1]=0 → _[3,2]=1 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=0 → _[4,4]=1 → [16]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=0 → _[2,4]=1 → _[3,1]=0 → _[3,2]=0 → _[3,3]=1 → _[3,4]=0 → _[4,1]=0 → _[4,2]=1 → _[4,3]=0 → _[4,4]=0 → [17]: → _[1,1]=1 → _[1,2]=0 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=0 → _[2,4]=1 → _[3,1]=0 → _[3,2]=1 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=1 → _[4,4]=0 → [18]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=1 → _[1,4]=0 → _[2,1]=1 → _[2,2]=0 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=1 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=0 → _[4,4]=1 → [19]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=1 → _[2,1]=1 → _[2,2]=0 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=1 → _[3,4]=0 → _[4,1]=0 → _[4,2]=1 → _[4,3]=0 → _[4,4]=0 → [20]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=0 → _[1,4]=0 → _[2,1]=1 → _[2,2]=0 → _[2,3]=0 → _[2,4]=0 → _[3,1]=0 → _[3,2]=0 → _[3,3]=0 → _[3,4]=1 → _[4,1]=0 → _[4,2]=0 → _[4,3]=1 → _[4,4]=0 → [21]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=1 → _[2,1]=0 → _[2,2]=0 → _[2,3]=1 → _[2,4]=0 → _[3,1]=0 → _[3,2]=1 → _[3,3]=0 → _[3,4]=0 → _[4,1]=1 → _[4,2]=0 → _[4,3]=0 → _[4,4]=0 → [22]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=0 → _[1,4]=1 → _[2,1]=0 → _[2,2]=0 → _[2,3]=1 → _[2,4]=0 → _[3,1]=1 → _[3,2]=0 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=1 → _[4,3]=0 → _[4,4]=0 → [23]: → _[1,1]=0 → _[1,2]=1 → _[1,3]=0 → _[1,4]=0 → _[2,1]=0 → _[2,2]=0 → _[2,3]=0 → _[2,4]=1 → _[3,1]=1 → _[3,2]=0 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=1 → _[4,4]=0 → [24]: → _[1,1]=0 → _[1,2]=0 → _[1,3]=1 → _[1,4]=0 → _[2,1]=0 → _[2,2]=1 → _[2,3]=0 → _[2,4]=0 → _[3,1]=1 → _[3,2]=0 → _[3,3]=0 → _[3,4]=0 → _[4,1]=0 → _[4,2]=0 → _[4,3]=0 → _[4,4]=1 ```

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#### 1.8 makeSymmetricGroup

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

makeSymmetricGroup(n[,k]);

Return:

list of matrices

Purpose:

compute matrix representation of symmetric group on n points

Note:

If k<>0 is given, the returned list contains only the two generators of the symmetric group. Otherwise and by default, all n! elements are computed.

Example:

 ```LIB "codessystem.lib"; ring r = 0,x,dp; list G = makeSymmetricGroup(4); printMatrixList(G); → [1]: → 0,0,0,1, → 1,0,0,0, → 0,1,0,0, → 0,0,1,0 → [2]: → 0,0,1,0, → 0,0,0,1, → 1,0,0,0, → 0,1,0,0 → [3]: → 1,0,0,0, → 0,0,0,1, → 0,1,0,0, → 0,0,1,0 → [4]: → 0,1,0,0, → 0,0,1,0, → 0,0,0,1, → 1,0,0,0 → [5]: → 0,0,0,1, → 0,0,1,0, → 1,0,0,0, → 0,1,0,0 → [6]: → 0,0,1,0, → 1,0,0,0, → 0,0,0,1, → 0,1,0,0 → [7]: → 1,0,0,0, → 0,1,0,0, → 0,0,1,0, → 0,0,0,1 → [8]: → 0,0,1,0, → 0,1,0,0, → 0,0,0,1, → 1,0,0,0 → [9]: → 0,1,0,0, → 0,0,0,1, → 0,0,1,0, → 1,0,0,0 → [10]: → 0,1,0,0, → 0,0,1,0, → 1,0,0,0, → 0,0,0,1 → [11]: → 1,0,0,0, → 0,0,1,0, → 0,0,0,1, → 0,1,0,0 → [12]: → 0,1,0,0, → 1,0,0,0, → 0,0,1,0, → 0,0,0,1 → [13]: → 1,0,0,0, → 0,0,1,0, → 0,1,0,0, → 0,0,0,1 → [14]: → 1,0,0,0, → 0,1,0,0, → 0,0,0,1, → 0,0,1,0 → [15]: → 0,0,0,1, → 0,1,0,0, → 0,0,1,0, → 1,0,0,0 → [16]: → 0,0,1,0, → 0,1,0,0, → 1,0,0,0, → 0,0,0,1 → [17]: → 0,0,0,1, → 0,1,0,0, → 1,0,0,0, → 0,0,1,0 → [18]: → 0,0,0,1, → 1,0,0,0, → 0,0,1,0, → 0,1,0,0 → [19]: → 0,0,1,0, → 1,0,0,0, → 0,1,0,0, → 0,0,0,1 → [20]: → 0,1,0,0, → 1,0,0,0, → 0,0,0,1, → 0,0,1,0 → [21]: → 0,0,0,1, → 0,0,1,0, → 0,1,0,0, → 1,0,0,0 → [22]: → 0,0,1,0, → 0,0,0,1, → 0,1,0,0, → 1,0,0,0 → [23]: → 0,1,0,0, → 0,0,0,1, → 1,0,0,0, → 0,0,1,0 → [24]: → 1,0,0,0, → 0,0,0,1, → 0,0,1,0, → 0,1,0,0 G = makeSymmetricGroup(5,1); printMatrixList(G); → [1]: → 0,0,0,0,1, → 1,0,0,0,0, → 0,1,0,0,0, → 0,0,1,0,0, → 0,0,0,1,0 → [2]: → 0,1,0,0,0, → 1,0,0,0,0, → 0,0,1,0,0, → 0,0,0,1,0, → 0,0,0,0,1 ```

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#### 1.9 inList

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

inList(m,L); m any type, L list

Return:

int, 0 or position of first appearance of m in L

Purpose:

check whether m is a member of L

Example:

 ```LIB "codessystem.lib"; list L = 1,2,3,4,5; inList(3,L); → 3 inList(6,L); → 0 ```

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#### 1.10 printMatrixList

Procedure from library `codessystem.lib` (see codessystem_lib).

Usage:

printMatrixList(L); L list of matrices

Return:

nothing

Purpose:

pretty-print of list of matrices

Example:

 ```LIB "codessystem.lib"; ring r = 0,x,dp; list G = makeSymmetricGroup(4); printMatrixList(G); → [1]: → 0,0,0,1, → 1,0,0,0, → 0,1,0,0, → 0,0,1,0 → [2]: → 0,0,1,0, → 0,0,0,1, → 1,0,0,0, → 0,1,0,0 → [3]: → 1,0,0,0, → 0,0,0,1, → 0,1,0,0, → 0,0,1,0 → [4]: → 0,1,0,0, → 0,0,1,0, → 0,0,0,1, → 1,0,0,0 → [5]: → 0,0,0,1, → 0,0,1,0, → 1,0,0,0, → 0,1,0,0 → [6]: → 0,0,1,0, → 1,0,0,0, → 0,0,0,1, → 0,1,0,0 → [7]: → 1,0,0,0, → 0,1,0,0, → 0,0,1,0, → 0,0,0,1 → [8]: → 0,0,1,0, → 0,1,0,0, → 0,0,0,1, → 1,0,0,0 → [9]: → 0,1,0,0, → 0,0,0,1, → 0,0,1,0, → 1,0,0,0 → [10]: → 0,1,0,0, → 0,0,1,0, → 1,0,0,0, → 0,0,0,1 → [11]: → 1,0,0,0, → 0,0,1,0, → 0,0,0,1, → 0,1,0,0 → [12]: → 0,1,0,0, → 1,0,0,0, → 0,0,1,0, → 0,0,0,1 → [13]: → 1,0,0,0, → 0,0,1,0, → 0,1,0,0, → 0,0,0,1 → [14]: → 1,0,0,0, → 0,1,0,0, → 0,0,0,1, → 0,0,1,0 → [15]: → 0,0,0,1, → 0,1,0,0, → 0,0,1,0, → 1,0,0,0 → [16]: → 0,0,1,0, → 0,1,0,0, → 1,0,0,0, → 0,0,0,1 → [17]: → 0,0,0,1, → 0,1,0,0, → 1,0,0,0, → 0,0,1,0 → [18]: → 0,0,0,1, → 1,0,0,0, → 0,0,1,0, → 0,1,0,0 → [19]: → 0,0,1,0, → 1,0,0,0, → 0,1,0,0, → 0,0,0,1 → [20]: → 0,1,0,0, → 1,0,0,0, → 0,0,0,1, → 0,0,1,0 → [21]: → 0,0,0,1, → 0,0,1,0, → 0,1,0,0, → 1,0,0,0 → [22]: → 0,0,1,0, → 0,0,0,1, → 0,1,0,0, → 1,0,0,0 → [23]: → 0,1,0,0, → 0,0,0,1, → 1,0,0,0, → 0,0,1,0 → [24]: → 1,0,0,0, → 0,0,0,1, → 0,0,1,0, → 0,1,0,0 ```

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# 2. Index

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where the Example assumes that the current position is at Subsubsection One-Two-Three of a document of the following structure:

• 1. Section One
• 1. Subsection One-One
• ..
• 1.2 Subsection One-Two
• 1.2.1 Subsubsection One-Two-One
• 1.2.2 Subsubsection One-Two-Two
• 1.2.3 Subsubsection One-Two-Three     <== Current Position
• 1.2.4 Subsubsection One-Two-Four
• 1.3 Subsection One-Three
• ..
• 1.4 Subsection One-Four

This document was generated by Daniel Andres on April, 15 2011 using texi2html 1.76.