Maximum distance separable (MDS) code

Maximum distance separable (MDS) code[1]

Description

A type of \(q\)-ary code whose parameters satisfy the Singleton bound with equality.

A \([n,k,d]_q\) \(q\)-ary linear code is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the Singleton bound \begin{align} d \leq n-k+1 \tag*{(1)}\end{align} becomes an equality. A code is called almost MDS (AMDS) when \(d=n-k\). A bound for general (i.e., nonlinear or unrestricted) \(q\)-ary codes can also be formulated; see [2; Thm. 1.9.10]. A code is near MDS (NMDS) if the code and its dual are mode AMDS.

The codes \( [n,1,n]_q, [n,n-1,2]_q, [n,n,1]_q \) for any \(q\) are MDS codes. These are called the trivial MDS codes. The only binary MDS codes are the trivial ones. Many, but not all, \(q\)-ary MDS codes are related to RS codes and their extensions; see, e.g., [3; Prob. 11.11].

Protection

Given \(n\) and \(k\), MDS codes have the highest distance possible of all codes and so have the best possible error-correction properties.

Realizations

Automatic repeat request (ARQ) data transmission protocols ([4], Ch. 7).

Notes

See Refs. [5][3; Sec. 11.4 notes][6; Ch. 11 notes] for more on MDS codes and the MDS conjecture.

Cousins

Dual linear code— A linear binary or \(q\)-ary \([n,k,n-k+1]\) code is MDS if and only if its dual \([n,n-k,k+1]\) is MDS [2; Thm. 1.9.13].
Projective geometry code— A linear code is MDS (almost MDS) if and only if columns of its parity-check matrix form an \(n\)-arc (\(n\)-track) in projective space [7–10]. The dual of a MDS code is an MDS code, so MDS codes are projective. All \([[9,3]]\) MDS codes have been tabulated [11] in terms of 9-arcs in the projective plane.
Distributed computation code— The first matrix multiplication code encoded each entry of the matrices to be multiplied into an MDS code [12].
MDS array code— MDS array codes are MDS codes when each matrix codeword is treated as a vector by converting each column into a single coordinate via subpacketization.
Maximum-rank distance (MRD) code— MRD codes are matrix-code analogues of MDS codes.
Maximum-sum-rank distance (MSRD) code
Algebraic-geometry (AG) code— Near MDS \([n,k,d]_{p^m}\) AG codes exist when \(n,p,m\) satisfy certain relations according to the Tsfasman-Vladut bound [13–15].
Elliptic code— Elliptic codes can be MDS [16; Exam. 15.5.3][14; pg. 310][13; Sec. 4.4.2].
Extended GRS code— A GRS code can be extended to an MDS code ([17], Thm. 5.3.4). Extended and doubly extended narrow-sense RS codes are MDS ([17], Thms. 5.3.2 and 5.3.4), and there is an equivalence between the two for odd prime \(q\) [18].
Narrow-sense RS code— Extended and doubly extended narrow-sense RS codes are MDS [17; Thms. 5.3.2 and 5.3.4], and there is an equivalence between the two for odd prime \(q\) [18].
Reed-Solomon (RS) code— If \(k \leq p\), then all linear MDS codes \( [n,k,n-k+1]_{p^m} \) are RS codes [18].
Generalized Srivastava code— Generalized Srivastava codes for \(m=1\) are MDS codes [6; pg. 359].
\(q\)-ary Hamming code— The \((4,9,3)_3\) Hamming code is a unique MDS code [19,20].
Quantum maximum-distance-separable (MDS) code
Perfect-tensor code— MDS codes can be used to obtain cluster states that are AME with minimal support [21–26].
Quantum tensor-product code— MDS codes can be used to construct quantum tensor-product codes [27].
EA MDS code— MDS codes give rise to families of EA Galois-qudit codes that saturate the original (erroneous) EAQECC Singleton bound [28].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Parents

Linear \(q\)-ary code

Optimal LRCLRC Distributed-storage ECC

The generalized Singleton bound becomes the Singleton bound for \(k=r\), so optimal LRCs with that constraint are MDS.

Universally optimal \(q\)-ary codeUniversally optimal ECC

MDS codes are LP universally optimal codes [29].

Orthogonal array (OA)\(t\)-design ECC

An MDS code is an OA\(_{1}(k,n,q)\) [30; Thm. 3.3.19].

Maximum distance separable (MDS) code

Children

Repetition code

Generalized RS (GRS) code

GRS codes have distance \(n-k+1\), saturating the Singleton bound.

Roth-Lempel code

Roth-Lempel codes are examples of MDS codes that are not GRS codes.

Glynn code

The Glynn code is a rare example of an MDS code that is not related to an RS code.

Hexacode

The hexacode is an MDS code [17; Exer. 578].

Hirschfeld code

The Hirschfeld code is a rare example of an MDS code that is not related to an RS code.

\(q\)-ary parity-check code

Tetracode

Griesmer code

Singleton bound implies the Griesmer bound.

References

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[2]: W. C. Huffman, J.-L. Kim, and P. Solé, "Basics of coding theory." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[3]: R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
[4]: S. B. Wicker and V. K. Bhargava, “Reed-Solomon Codes and Their Applications”, (1999) DOI
[5]: J. W. P. Hirschfeld and J. A. Thas, “Open problems in finite projective spaces”, Finite Fields and Their Applications 32, 44 (2015) DOI
[6]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[7]: R. C. Bose (1947). Mathematical theory of the symmetrical factorial design. Sankhyā: The Indian Journal of Statistics, 107-166.
[8]: S. M. Dodunekov and I. N. Landgev, “On near-MDS codes”, Proceedings of 1994 IEEE International Symposium on Information Theory 427 DOI
[9]: J. W. P. Hirschfeld and L. Storme, “The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001”, Developments in Mathematics 201 (2001) DOI
[10]: Beutelspacher, Albrecht, and Ute Rosenbaum. Projective geometry: from foundations to applications. Cambridge University Press, 1998.
[11]: A. V. Iampolskaia, A. N. Skorobogatov, and E. A. Sorokin, “Formula for the number of [9,3] MDS codes”, IEEE Transactions on Information Theory 41, 1667 (1995) DOI
[12]: K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, “Speeding Up Distributed Machine Learning Using Codes”, IEEE Transactions on Information Theory 64, 1514 (2018) arXiv:1512.02673 DOI
[13]: M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.
[14]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[15]: I. N. Landjev, “Linear codes over finite fields and finite projective geometries”, Discrete Mathematics 213, 211 (2000) DOI
[16]: A. Couvreur, H. Randriambololona, "Algebraic Geometry Codes and Some Applications." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[17]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[18]: S. Ball, “On sets of vectors of a finite vector space in which every subset of basis size is a basis”, Journal of the European Mathematical Society 14, 733 (2012) DOI
[19]: Taussky, Olga, and John Todd. "Covering theorems for groups." Bulletin of the American Mathematical Society. Vol. 54. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1948.
[20]: J. G. Kalbfleisch and R. G. Stanton, “A Combinatorial Problem in Matching”, Journal of the London Mathematical Society s1-44, 60 (1969) DOI
[21]: A. V. Thapliyal, Multipartite maximally entangled states, minimal entanglement generating states and entropic inequalities unpublished presentation (2003).
[22]: W. Helwig and W. Cui, “Absolutely Maximally Entangled States: Existence and Applications”, (2013) arXiv:1306.2536
[23]: W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
[24]: D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, “Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices”, Physical Review A 92, (2015) arXiv:1506.08857 DOI
[25]: Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
[26]: D. Alsina, “PhD thesis: Multipartite entanglement and quantum algorithms”, (2017) arXiv:1706.08318
[27]: J. Fan, Y. Li, M.-H. Hsieh, and H. Chen, “On Quantum Tensor Product Codes”, (2017) arXiv:1605.09598
[28]: J. Fan, H. Chen, and J. Xu, “Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q + 1”, (2016) arXiv:1602.02235
[29]: H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
[30]: P. R. J. Östergård, "Construction and Classification of Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI

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Victor V. Albert (2024-07-11)
Victor V. Albert (2022-08-09)
Victor V. Albert (2022-04-28)
Victor V. Albert (2021-12-16)
Eric Kubischta (2021-12-15)

Cite as:

“Maximum distance separable (MDS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/mds

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