Description
An \([n,n/2]_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to an inner product, most commonly either Euclidean or Hermitian. Self-dual codes exist only for even lengths and have dimension \(k=n/2\). A code that is equivalent to its dual is called iso-dual.
An even (doubly even) self-dual code is called Type I (Type II) [1,2]. Ternary (quaternary) self-dual codes are called Type III (Type IV), and each of their codewords has weight three (two).
Protection
The generator matrix of the Hermitian dual of a code with generator matrix \(G = [I_k~~A]\) is \([-\bar{A}^T~~I_{n-k}]\), where \(\bar{A}\) contains matrix elements of \(A\) raised to the \(p\)th power. A code is Hermitian self-dual if and only if \(A \bar{A}^{T} = -I_{n/2}\).
The minimum distance of a Hermitian self-dual \([n,n/2]\) code satisfies \begin{align} d\leq\begin{cases} 2\left\lfloor \frac{n}{8}\right\rfloor +2 & q=2\text{ and code is singly-even}\\ 4\left\lfloor \frac{n}{24}\right\rfloor +4 & q=2\text{ and code is doubly even}\\ 3\left\lfloor \frac{n}{12}\right\rfloor +3 & q=3\\ 2\left\lfloor \frac{n}{6}\right\rfloor +2 & q=4\text{ and code is even} \end{cases}~, \tag*{(1)}\end{align} except for \(n = 22\) modulo four for the second case, where the bound is increased by four [3]. A self-dual code saturating the above inequality is called extremal.
Fixed-weight codewords of extremal self-dual doubly even binary codes whose length divides 24 form a combinatorial 5-design [4]. The extended Golay code and the \([48,24,12]\) self-dual code are two such codes. It is not yet known whether a \([72,36,16]\) self-dual code exists; see [5,6][7; Remark 4.3.11].
For ternary self-dual codes, see [8][7; Remark 4.3.14].
Notes
See books [9,10] for more on self-dual codes.See Refs. [11,12] for constructions of binary self-dual codes.See Tables of Self-Dual Codes for a database of self-dual codes over \(GF(2)\), \(GF(3)\), \(GF(4)\) (Euclidean or Hermitian), \(GF(5)\), and \(GF(7)\). See also Ref. [13].Cousins
- Divisible code— Binary self-dual codes are singly-even, and binary self-orthogonal codes that are not doubly even are singly-even [14; Def. 4.1.6]. The minimum distance of doubly even binary self-dual codes asymptotically satisfies \(d\leq0.1664n+o(n)\) [15].
- Unimodular lattice— Unimodular lattices are lattice analogues of self-dual codes. There are several parallels between (doubly even) self-dual binary codes and (even) unimodular lattices [2,9].
- Niemeier lattice— Niemeier lattices can be constructed from ternary self-dual codes of length 24 [16].
- Combinatorial design— Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [4] (see [17; Sec. 5.4]). See [18; Table 1.61, pg. 683] for a table of combinatorial designs obtained from self-dual codes.
- Nordstrom-Robinson (NR) code— The NR code is self-dual in that its distance distribution is invariant under the MacWilliams transform [19]. It maps to the octacode, a self-dual code over \(\mathbb{Z}_4\) under the Gray map [20,21].
- Ternary Golay code— The extended ternary Golay code is self-dual.
- Cyclic linear \(q\)-ary code— See Refs. [22,23] for tables of cyclic self-dual codes.
- Jump code— Iso-dual codes can be used to construct jump codes [24].
- Hermitian qubit code— Hermitian qubit codes are constructed from Hermitian self-orthogonal linear codes over \(GF(4)\) via the \(GF(4)\) representation. This relation yields bounds on self-dual codes over \(GF(4)\) [25].
- Triorthogonal code— Self-dual binary codes can be used to construct triorthogonal codes [26].
Primary Hierarchy
References
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Page edit log
- Victor V. Albert (2023-04-21) — most recent
Cite as:
“Self-dual linear code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/self_dual