A \([n,k]\)-Linear code over the field f. The code parameters f,n and k are carried on the type level. A linear code is a subspace C of \(f^n\) generated by the generator matrix.

A Generator is the generator matrix of a linear code, not necessarily in standard form.

A CheckMatrix or parity matrix is the dual of a Generator. It can be used to check if a word is a valid code word for the code. Also, \[ \forall v \in f^k: cG \cdot H^\top = 0 \] i.e. the code is generated by the kernel of a check matrix.

Generate a linear \( [n,k]_q \)-Code over the field f with the generator in standard form (I|A), where the given function generates the \( k \times (n-k) \)-matrix A. The distance is unknown for this code and thus decoding algorithms may be very inefficient.

Generate a linear \( [n,k,d]_q \)-Code over the field f with the generator in standard form (I|A), where the given function generates the \( k \times (n-k) \)-matrix A.

Uses Gaussian eleminiation via rref from Matrix to find the standard form of generators.

The standard from generator of a linear code. Uses standardForm to calculate a standard form generator.

A row vector (a matrix with one row).

Get the codeword generated by the given k-sized vector.

Check if the given candidate code word is a valid code word for the given linear code. If not, the party check failed.

Check if the given candidate code word has errors, i.e. if some element changed during transmission. This is equivalent with not isCodeword

The hamming weight of a Vector is an Int between 0 and n

A list of all codewords

List all vectors of length n over field f

List all vectors of length n with non-zero elements over field f

List of all words with given hamming weight

List of all words with (non-zero) hamming weight smaller than a given boundary

Give the syndrome of a word for the given code. This is 0 if the word is a valid code word.

Synonym for syndromeDecoding, an inefficient decoding algorithm that works for all linear codes.

Uses the exponential-time syndrome decoding algorithm for general codes. c.f: https://en.wikipedia.org/wiki/Decoding_methods#Syndrome_decoding

Return a syndrome table for the given linear code. If the distance is not known (i.e. minDist c = Nothing) this is very inefficient.

Replace the syndromeTable of a code with a newly calculated syndrome table for the (current) generator. Useful to get a syndrome table for transformed codes when the table cannot be transformed, too.

A syndrome table is a map from syndromes to their minimal weight representative. Every vector v has a syndrome \( S(v) \). This table reverses the syndrome function S and chooses the vector with the smallest hamming weight from it's image. This is a lookup table for syndrome decoding.

The dual code is the code generated by the check matrix

This drops already calculated syndromeTables.

The dual code is the code generated by the check matrix.

This drops already calculated syndromeTables.

Permute the rows of a code with a permutation matrix. The given permutation matrix must be a valid permutation matrix; this is not checked. This effectively multiplies the generator and check matrix from the right. Te distance of the resulting code stays the same.

This drops already calculated syndromeTables.

Extend the given code \( c \) by zero-columns. Vectors \( v_{ext} \in c_{ext} \) have the form \( v = (v_1, \dots , v_n, 0, \dots, 0) \) . The distance of the extended code stays the same. This drops a calculated syndromeTable and makes it necessary to recalculate it if it's accessed.

The trivial code is the identity code where the parity bits are all zero.

A simplex code is a code generated by all possible codewords consisting of 0's and 1's except the zero vector.

The Hamming(7,4)-code. It is a [7,4,3]_2 code

The _Golay_-code is a perfect [24,12,7]-code. It is the only other non-trivial perfect code and the only perfect code that is able to correct 3 errors.

Syndrome decoding on this code takes a very, very long time.

A binary code is a linear code over the field GF(2)

A random permutation matrix

Convenience function to extract the length n from the type level

Convenience function to extract the rank k from the type level.

Standard base vector [0..0,1,0..0] for any field. Parameter must be >=1

First base vector [1,0..0]

Second base vector [0,1,0..0]

Characteristic of a field. It takes a finite field type in the proxy value and gives the characteristic. This is done using type families To support new finite field types, you need to add a type instance for the type family Characteristic.

O(rows*cols). Generate a matrix from a generator function. | The elements are 1-indexed, i.e. top-left element is (1,1).

O(rows*cols). The zero matrix

O(rows*cols). The transpose of a matrix.

Create a matrix from a list of elements. The list must have exactly length n*m. This is checked or else an exception is thrown.

Create a matrix from a list of rows. The list must have exactly m lists of length n. An exception is thrown otherwise.