Index of code constructions

The codes object may be used to access the codes that Sage can build.

DuadicCodeEvenPair() Constructs the “even pair” of duadic codes associated to the “splitting” (see the docstring for _is_a_splitting for the definition) S1, S2 of n.
DuadicCodeOddPair() Constructs the “odd pair” of duadic codes associated to the “splitting” S1, S2 of n.
ExtendedQuadraticResidueCode() The extended quadratic residue code (or XQR code) is obtained from a QR code by adding a check bit to the last coordinate. (These codes have very remarkable properties such as large automorphism groups and duality properties - see [HP2003], Section 6.6.3-6.6.4.)
QuadraticResidueCode() A quadratic residue code (or QR code) is a cyclic code whose generator polynomial is the product of the polynomials \(x-\alpha^i\) (\(\alpha\) is a primitive \(n^{th}\) root of unity; \(i\) ranges over the set of quadratic residues modulo \(n\)).
QuadraticResidueCodeEvenPair() Quadratic residue codes of a given odd prime length and base ring either don’t exist at all or occur as 4-tuples - a pair of “odd-like” codes and a pair of “even-like” codes. If \(n > 2\) is prime then (Theorem 6.6.2 in [HP2003]) a QR code exists over \(GF(q)\) iff q is a quadratic residue mod \(n\).
QuadraticResidueCodeOddPair() Quadratic residue codes of a given odd prime length and base ring either don’t exist at all or occur as 4-tuples - a pair of “odd-like” codes and a pair of “even-like” codes. If n 2 is prime then (Theorem 6.6.2 in [HP2003]) a QR code exists over GF(q) iff q is a quadratic residue mod n.
QuasiQuadraticResidueCode() A (binary) quasi-quadratic residue code (or QQR code).
RandomLinearCodeGuava() The method used is to first construct a \(k \times n\) matrix of the block form \((I,A)\), where \(I\) is a \(k \times k\) identity matrix and \(A\) is a \(k \times (n-k)\) matrix constructed using random elements of \(F\). Then the columns are permuted using a randomly selected element of the symmetric group \(S_n\).
ReedMullerCode() Returns a Reed-Muller code.
ReedSolomonCode() Construct a classical Reed-Solomon code.
ToricCode() Let \(P\) denote a list of lattice points in \(\ZZ^d\) and let \(T\) denote the set of all points in \((F^x)^d\) (ordered in some fixed way). Put \(n=|T|\) and let \(k\) denote the dimension of the vector space of functions \(V = \mathrm{Span}\{x^e \ |\ e \in P\}\). The associated toric code \(C\) is the evaluation code which is the image of the evaluation map
WalshCode() Return the binary Walsh code of length \(2^m\).
from_parity_check_matrix() Return the linear code that has H as a parity check matrix.
random_linear_code() Generate a random linear code of length length, dimension dimension and over the field F.

Note

To import these names into the global namespace, use:

sage: from sage.coding.codes_catalog import *