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 via permutations 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
Parents
- Dual linear code
- Self-dual additive code — Self-dual linear codes with respect to some inner product are automatically self-dual additive under the same inner product since linear codes are additive. In addition, quaternary linear codes are Hermitian self-orthogonal (self-dual) iff they are trace-Hermitian self-orthogonal (self-dual) additive [14; Thm. 27.4.1] ([10; Thm. 9.10.3]).
Children
- \([48,24,12]\) self-dual code — The \([48,24,12]\) self-dual code is the only self-dual doubly-even code at its parameters [15].
- \([8,4,4]\) extended Hamming code — The \([8,4,4]\) extended Hamming code is the smallest doubly-even self-dual code.
- Pless symmetry code
- Hexacode — The hexacode is Hermitian self-dual and, as a result, is also trace-Hermitian self-dual additive [10; Sec. 9.10]. The hexacode and the shortened hexacode are extremal [10; Tab. 9.14][16; Tm. 12].
- Tetracode — The tetracode is Euclidean self-dual.
Cousins
- Divisible code — Binary self-dual codes are singly-even, and binary self-orthogonal codes that are not doubly-even are singly-even [17; Def. 4.1.6]. The minimum distance of doubly-even binary self-dual codes asymptotically satisfies \(d\leq0.1664n+o(n)\) [18].
- Unimodular lattice code — 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 code — Niemeier lattice codes can be constructed from ternary self-dual codes of length 24 [19].
- Combinatorial design — Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [4] (see [20; Sec. 5.4]). See [21; Table 1.61, pg. 683] for a table of combinatorial designs obtained from self-dual codes.
- Golay code — The extended Golay code is the unique code at its parameters and happens to be self-dual [22][7; Remark 4.3.11].
- Nordstrom-Robinson (NR) code — The NR code is self-dual in that its distance distribution is invariant under the MacWilliams transform [23]. It maps to the octacode, a self-dual code over \(\mathbb{Z}_4\) under the Gray map [24,25].
- Ternary Golay code — The extended ternary Golay code is self-dual.
- Cyclic linear \(q\)-ary code — See Refs. [26,27] for tables of cyclic self-dual codes.
- Jump code — Iso-dual codes can be used to construct jump codes [28].
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