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
Realizations
Notes
Parents
- Linear \(q\)-ary code
- Optimal LRC — The generalized Singleton bound becomes the Singleton bound for \(k=r\), so optimal LRCs with that constraint are MDS.
- Universally optimal \(q\)-ary code — MDS codes are LP universally optimal codes [7].
- Orthogonal array (OA) — An MDS code is an OA\(_{1}(k,n,q)\) [8; Thm. 3.3.19].
Children
- 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 [9; 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.
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 [10–13]. The dual of a MDS code is an MDS code, so MDS codes are projective. All \([[9,3]]\) MDS codes have been tabulated [14] 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 [15].
- 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 [16–18].
- Elliptic code — Elliptic codes can be MDS [19; Exam. 15.5.3][17; pg. 310][16; Sec. 4.4.2].
- Extended GRS code — A GRS code can be extended to an MDS code ([9], Thm. 5.3.4). Extended and doubly extended narrow-sense RS codes are MDS ([9], Thms. 5.3.2 and 5.3.4), and there is an equivalence between the two for odd prime \(q\) [20].
- Narrow-sense RS code — Extended and doubly extended narrow-sense RS codes are MDS [9; Thms. 5.3.2 and 5.3.4], and there is an equivalence between the two for odd prime \(q\) [20].
- Reed-Solomon (RS) code — If \(k \leq p\), then all linear MDS codes \( [n,k,n-k+1]_{p^m} \) are RS codes [20].
- 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 [21,22].
- Quantum maximum-distance-separable (MDS) code
- Perfect-tensor code — MDS codes can be used to obtain perfect-tensor codes with minimal support [23–25].
- Quantum tensor-product code — MDS codes can be used to construct quantum tensor-product codes [26].
- EA MDS code — MDS codes give rise to families of EA Galois-qudit codes that saturate the original (erroneous) EAQECC Singleton bound [27].
References
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- 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
Page edit log
- Markus Grassl (2024-07-11) — most recent
- 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