Online manual of codessystem.lib

1. codessystem_lib  
2. Index  
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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:

1.1 hammingWeightEnumerator   Hamming weight enumerator of code C
1.2 completeWeightEnumerator   complete weight enumerator of code C
1.3 hammingWeight   Hamming weight
1.4 dualCode   compute the dual code of C
1.5 isSelfDual   check whether a linear code is self-dual
1.6 completeCode   all elements of linear space spanned by rows of C

Auxiliary procedures:

1.7 simpleOrbits   simple orbit algorithm
1.8 makeSymmetricGroup   creates matrix repr. of symmetric group on n points
1.9 inList   check whether m is in list L
1.10 printMatrixList   pretty-print of list of matrices

See also: finvar_lib.

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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
[ < ] [ > ]   [ << ] [ Up ] [ >> ]         [Top] [Index] [ ? ]

2. Index

Jump to:   C   D   H   I   M   P   S  
Index Entry Section
C
codessystem.lib1. codessystem_lib
codessystem_lib1. codessystem_lib
completeCode1.6 completeCode
completeWeightEnumerator1.2 completeWeightEnumerator
D
dualCode1.4 dualCode
H
hammingWeight1.3 hammingWeight
hammingWeightEnumerator1.1 hammingWeightEnumerator
I
inList1.9 inList
isSelfDual1.5 isSelfDual
M
makeSymmetricGroup1.8 makeSymmetricGroup
P
printMatrixList1.10 printMatrixList
S
simpleOrbits1.7 simpleOrbits
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