Linear \(q\)-ary code 


An \((n,K,d)_q\) linear code is denoted as \([n,k,d]_q\), where \(k=\log_q K\) need not be an integer. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(\alpha x+ \beta y\) is also a codeword for any \(q\)-ary digits \(\alpha,\beta\). This extra structure yields much information about their properties, making them a large and well-studied subset of codes.

Linear codes can be defined in terms of a generator matrix \(G\), whose rows form a basis for the \(k\)-dimensional codespace. Given a message \(x\), the corresponding encoded codeword is \(G^T x\). The generator matrix can be reduced via coordinate permutations to its standard or systematic form \(G = [I_k~~A]\), where \(I_k\) is a \(k\times k\) identity matrix and \(A\) is a \(k \times (n-k)\) \(q\)-ary matrix.

The two extreme cases are the \([n,0,n]\) zero code and its dual the \([n,n,1]\) universe code.


Distance \(d\) of a linear code is the number of nonzero entries in the (nonzero) codeword with the smallest such number. Corrects any error set such that the difference of any pair of distinct elements of the set is a codeword.

Geometrically local \(q\)-ary codes are limited by the classical Bravyi-Poulin-Terhal (BPT) bound [1], known to be tight in any Euclidean dimension [2].


Maximum likelihood (ML) decoding. This algorithm decodes a received word to the most likely sent codeword based on the received word. ML decoding of reduced complexity is possible for virtually all \(q\)-ary linear codes [3].Optimal symbol-by-symbol decoding rule [4].Information set decoding (ISD) [5], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Generalized minimum-distance decoder [6].Soft-decision maximum-likelihood trellis-based decoder [7].Random linear codes over large fields are list-recoverable and list-decodable up to near-optimal rates [8].Extensions of algebraic-geometry decoders to linear codes [9,10].


Admits a parity check matrix \(H\), whose columns make up a maximal linearly independent set of vectors that are in the kernel of \(G\).University of Salzburg's MinT application generates an optimal parameter table for a linear code \([n,k,d]_q\), contingent on an optional fluctuation of maximal Hamming code distance, rank, and length, along with other specifications.


  • Additive \(q\)-ary code — For \(q>2\), additive codes need not be linear since linearity also requires closure under multiplication.
  • \(R\)-linear code — Linear \(q\)-ary codes are \(GF(q)\)-linear.



  • \(q\)-ary linear LTC — Linear \(q\)-ary codes with distances \(\frac{1}{2}n-\sqrt{t n}\) for some \(t\) are called almost-orthogonal and are locally testable with query complexity of order \(O(t)\) [15]. This was later improved to codes with distance \(\frac{1}{2}n-O(n^{1-\gamma})\) for any positive \(\gamma\) [16], provided that the number of codewords is polynomial in \(n\).
  • Gabidulin code — Gabidulin codes over \(GF(q^N)\), when expressed as vectors over \(GF(q^N)\), are linear \(q\)-ary codes.
  • Locally recoverable code (LRC) — A \(q\)-ary linear code is an LRC of locality \(r\) if each coordinate participates in at least one parity check of weight \(\leq r\) [17].
  • Evaluation AG code — The degree of the divisor for evaluation AG codes is restricted to be less than \(n\). When there is no restriction, any \(q\)-ary linear code can be formulated as an evaluation AG code [18].
  • Entanglement-assisted (EA) QECC — Any linear \(q\)-ary code can be used to construct an EAQECC.
  • EA qubit stabilizer code — Any linear quaternary (\(q=4\)) code can be used to construct an EA qubit stabilizer code.
  • Galois-qudit CSS code — The Galois-qudit CSS construction uses two related \(q\)-ary linear codes, \(C_X\) and \(C_Z\).
  • True Galois-qudit stabilizer code — A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the symplectic representation of the stabilizers form a linear subspace.


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“Linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
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“Linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.