Operations on Codewords

Construction of a Codeword

C ! [a₁, ..., aₙ]: Code, [ RngElt ] -> ModTupRngElt

elt< C | a₁, ..., aₙ>: Code, List -> ModTupRngElt

Given a length \(n\) additive code \(C\) with alphabet \(F\), then the codewords of \(C\) lie in \(F^{(n)}\). Given elements \(a_1, \ldots, a_n\) belonging to \(F\), construct the codeword \((a_1, \ldots, a_n)\) of \(C\). A check is made that the vector (\(a_1, \ldots, a_n\)) is an element of \(C\).

C ! u: Code, ModTupRngElt -> ModTupRngElt

Given an additive code \(C\) which is defined as a subset of the \(F\)-space \(V = F^{(n)}\), and an element \(u\) belonging to \(V\), create the codeword of \(C\) corresponding to \(u\). The function will fail if \(u\) does not belong to \(C\).

C ! 0: Code, RngIntElt -> ModTupRngElt

The zero word of the additive code \(C\).

Random(C): Code -> ModTupRngElt

A random codeword of the additive code \(C\).

Arithmetic Operations on Codewords

u + v: ModTupRngElt, ModTupRngElt -> ModTupRngElt

Sum of the codewords \(u\) and \(v\), where \(u\) and \(v\) belong to the same linear code \(C\).

- u: ModTupRngElt -> ModTupRngElt

Additive inverse of the codeword \(u\) belonging to the linear code \(C\).

u - v: ModTupRngElt, ModTupRngElt -> ModTupRngElt

Difference of the codewords \(u\) and \(v\), where \(u\) and \(v\) belong to the same linear code \(C\).

a * u: RngElt, ModTupRngElt -> ModTupRngElt

Given an element \(a\) belonging to the alphabet \(F\), and a codeword \(u\) belonging to the additive code \(C\), return the codeword \(a*u\).

Normalize(u): ModTupRngElt -> ModTupRngElt

Normalize a codeword \(u\) of an additive code \(C\), returning a scalar multiple of \(u\) such that its first non-zero entry is \(1\).

Distance and Weight

Distance(u, v): ModTupRngElt, ModTupRngElt -> RngIntElt

The Hamming distance between the codewords \(u\) and \(v\), where \(u\) and \(v\) belong to the same additive code \(C\).

Weight(u): ModTupRngElt -> RngIntElt

The Hamming weight of the codeword \(u\), i.e., the number of non-zero components of \(u\).

Vector Space and Related Operations

(u, v): ModTupRngElt, ModTupRngElt -> RngElt

InnerProduct(u, v): ModTupRngElt, ModTupRngElt -> RngElt

Inner product of the vectors \(u\) and \(v\) with respect to the Euclidean norm, where \(u\) and \(v\) belong to the parent vector space of the code \(C\).

TraceInnerProduct(K, u, v): FldFin, ModTupFldElt, ModTupFldElt -> FldFinElt

Given vectors \(u\) and \(v\) defined over a finite field \(L\) and a subfield \(K\) of \(L\), this function returns the trace of the inner product of the vectors \(u\) and \(v\) with respect to \(K\).

Support(w): ModTupRngElt -> { RngIntElt }

Given a word \(w\) belonging to the \([n, k]\) code \(C\), return its support as a subset of the integer set \(\{ 1 .. n \}\). The support of \(w\) consists of the coordinates at which \(w\) has non-zero entries.

Coordinates(C, u): Code, ModTupRngElt -> [ RngFinElt ]

Given an \([n, k\, :\,k_g]\) \(K\)-additive code \(C\) and a codeword \(u\) of \(C\) return the coordinates of \(u\) with respect to the current basis of \(C\). The coordinates of \(u\) are returned as a sequence \(Q = [a_1, \ldots, a_{k_g}]\) of elements from \(K\) such that \(u = a_1 * C.1 + \ldots + a_{k_g} * C.{k_g}\).

Parent(w): ModTupRngElt -> ModTupRng

Given a word \(w\) belonging to the code \(C\), return the ambient space \(V\) of \(C\).

Rotate(u, k): ModTupRngElt, RngIntElt -> ModTupRngElt

Given a vector \(u\), return the vector obtained from \(u\) by cyclically shifting its components to the right by \(k\) coordinate positions.

Rotate(~u, k): ModTupRngElt, RngIntElt

Given a vector \(u\), destructively rotate \(u\) by \(k\) coordinate positions.

Trace(u, S): ModTupFldElt, FldFin -> ModTupFldElt

Trace(u): ModTupFldElt -> ModTupFldElt

Given a vector \(u\) with components in \(K\), and a subfield \(S\) of \(K\), construct the vector with components in \(S\) obtained from \(u\) by taking the trace of each component with respect to \(S\). If \(S\) is omitted, it is taken to be the prime field of \(K\).

Predicates for Codewords

u eq v: ModTupRngElt, ModTupRngElt -> BoolElt

The function returns true if and only if the codewords \(u\) and \(v\) belonging to the same additive code are equal.

u ne v: ModTupRngElt, ModTupRngElt -> BoolElt

The function returns true if and only if the codewords \(u\) and \(v\) belonging to the same additive code are not equal.

IsZero(u): ModTupRngElt -> BoolElt

The function returns true if and only if the codeword \(u\) is the zero vector.

Accessing Components of a Codeword

u[i]: ModTupRngElt, RngIntElt -> RngElt

Given a codeword \(u\) belonging to the code \(C\) defined over the ring \(R\), return the \(i\)-th component of \(u\) (as an element of \(R\)).

u[i] := x;

Given an element \(u\) belonging to a subcode \(C\) of the full \(R\)-space \(V = R^n\), a positive integer \(i\), \(1 \leq i\leq n\), and an element \(x\) of \(R\), this function returns a vector in \(V\) which is \(u\) with its \(i\)-th component redefined to be \(x\).