## 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

## Notes

## 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

- Linear binary code — Linear binary codes are linear \(q\)-ary codes for \(q=2\).
- Evaluation code — Evaluation codes are defined using polynomial or rational functions evaluated on a subset of affine or projective space. Given access to more general structures (i.e., morphisms of algebras), any \(q\)-ary linear code can be formulated as an evaluation code ([11], Sec. 4.1; [12], Prop. 1.1.4).
- Multiplicity code
- Folded RS (FRS) code — Folding an RS code yields a code that is no longer linear over \(GF(q)\) but is linear over \(GF(q^m)\).
- Interleaved RS (IRS) code — IRS codes are linear over \(GF(q)\) but not necessarily over \(GF(q^t)\).
- Maximum distance separable (MDS) code
- Roth-Lempel code
- Dual linear code
- \(q\)-ary Hamming code
- Quasi group-algebra code — A linear code is a quasi group-algebra code for a group \(G\) and index \(\ell\) if and only if \(G\) is isomorphic to a regular subgroup of the code's permutation automorphism group which acts freely of index \(\ell\) on the coordinates [13; Thm. 3.5].
- \(q\)-ary linear LTC
- Projective geometry code — Columns of the generator matrix of a projective linear \([n,k]_q\) code correspond to distinct nonzero points in projective space. In general, linear codes admit repeating columns or columns proportional to each other. In that case, the columns correspond to a multiset of non-distinct nonzero points, and multisets are in one-to-one correspondence to arcs in projective space ([14], Thm. 1.1).
- Classical fractal liquid code
- \(q\)-ary LDGM code
- Tanner code
- Divisible code
- Two-weight code
- Wozencraft ensemble code

## 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.
- 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 [17].
- 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.
- Modular-qudit CSS code — Construction for prime \(q=p\) uses two related \(p\)-ary linear codes \(C_X\) and \(C_Z\).
- Galois-qudit CSS code — 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

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- 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.
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- R. Pellikaan, “On decoding by error location and dependent sets of error positions”, Discrete Mathematics 106–107, 369 (1992) DOI
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- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
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- M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.
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- M. Borello and W. Willems, “On the algebraic structure of quasi group codes”, (2021) arXiv:1912.09167
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- I. N. Landjev, “The Geometric Approach to Linear Codes”, Developments in Mathematics 247 (2001) DOI
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- T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) DOI
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- 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
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- 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

## Page edit log

- Victor V. Albert (2022-08-04) — most recent
- Micah Shaw (2022-06-08)
- Victor V. Albert (2022-03-21)
- Victor V. Albert (2021-10-30)

## Cite as:

“Linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_linear