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 *