Linear \(q\)-ary code 

Description

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.

Protection

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].

Decoding

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].

Notes

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.

Parents

  • 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.

Children

Cousins

  • \(q\)-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.

References

[1]
S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010) arXiv:0909.5200 DOI
[2]
N. Baspin, “On combinatorial structures in linear codes”, (2023) arXiv:2309.16411
[3]
I. Dumer, “Maximum likelihood decoding with reduced complexity”, Proceedings of IEEE International Symposium on Information Theory DOI
[4]
C. Hartmann and L. Rudolph, “An optimum symbol-by-symbol decoding rule for linear codes”, IEEE Transactions on Information Theory 22, 514 (1976) DOI
[5]
C. Peters, “Information-Set Decoding for Linear Codes over F q”, Post-Quantum Cryptography 81 (2010) DOI
[6]
G. Forney, “Generalized minimum distance decoding”, IEEE Transactions on Information Theory 12, 125 (1966) DOI
[7]
J. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis”, IEEE Transactions on Information Theory 24, 76 (1978) DOI
[8]
A. Rudra and M. Wootters, “Average-radius list-recovery of random linear codes: it really ties the room together”, (2017) arXiv:1704.02420
[9]
R. Kotter. A unified description of an error locating procedure for linear codes. In D. Yorgov, editor, Proc. 3rd International Workshop on Algebraic and Combinatorial Coding Theory, pages 113–117, Voneshta Voda, Bulgaria, June 1992. Hermes.
[10]
R. Pellikaan, “On decoding by error location and dependent sets of error positions”, Discrete Mathematics 106–107, 369 (1992) DOI
[11]
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[12]
M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.
[13]
M. Borello and W. Willems, “On the algebraic structure of quasi group codes”, (2021) arXiv:1912.09167
[14]
I. N. Landjev, “The Geometric Approach to Linear Codes”, Developments in Mathematics 247 (2001) DOI
[15]
T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) DOI
[16]
T. Kaufman and M. Sudan, “Sparse Random Linear Codes are Locally Decodable and Testable”, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07) (2007) DOI
[17]
L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653
[18]
R. Pellikan, B.-Z. Shen, and G. J. M. van Wee, “Which linear codes are algebraic-geometric?”, IEEE Transactions on Information Theory 37, 583 (1991) DOI
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Zoo Code ID: q-ary_linear

Cite as:
“Linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_linear
BibTeX:
@incollection{eczoo_q-ary_linear, title={Linear \(q\)-ary code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/q-ary_linear} }
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“Linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_linear

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/q-ary_linear.yml.