# 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 \(\alpha,\beta\).

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.

## Protection

## Decoding

## Notes

## Parents

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

## Children

- Cyclic linear \(q\)-ary code
- 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 [5].
- Folded RS (FRS) code
- Gabidulin code
- Generalized RM (GRM) code
- Interleaved RS (IRS) code — IRS codes are linear over \(GF(q)\) but not necessarily over \(GF(q^t)\).
- 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 ([6], Thm. 1.1).
- Wozencraft ensemble code
- \(q\)-ary Hamming code
- \(q\)-ary linear LTC

## 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 \(O(t)\) [7]. This was later improved to codes with distance \(\frac{1}{2}n-O(n^{1-\gamma})\) for any positive \(\gamma\) [8], provided that the number of codewords is polynomial in \(n\).
- Entanglement-assisted (EA) QECC — Any linear \(q\)-ary code can be used to construct an EAQECC.
- Entanglement-assisted (EA) stabilizer code — Any linear quaternary (\(q=4\)) code can be used to construct an EA stabilizer code.
- 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 ([9], Sec. 4.1; [10], Prop. 1.1.4).
- Galois-qudit CSS code — Construction uses two related \(q\)-ary linear codes \(C_X\) and \(C_Z\).
- Modular-qudit CSS code — Construction for prime \(q=p\) uses two related \(p\)-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]
- J. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis”, IEEE Transactions on Information Theory 24, 76 (1978). DOI
- [2]
- Atri Rudra and Mary Wootters, “Average-radius list-recovery of random linear codes: it really ties the room together”. 1704.02420
- [3]
- 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.
- [4]
- R. Pellikaan, “On decoding by error location and dependent sets of error positions”, Discrete Mathematics 106-107, 369 (1992). DOI
- [5]
- 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
- [6]
- I. N. Landjev, “The Geometric Approach to Linear Codes”, Developments in Mathematics 247 (2001). DOI
- [7]
- T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05). DOI
- [8]
- 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
- [9]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
- [10]
- M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.

## 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)

## Zoo code information

## Cite as:

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