Dual linear code

Cousins

• Dual lattice— Dual lattices are lattice analogues of dual codes.
• Combinatorial design— Linear codes and their duals are related to combinatorial designs via the Assmus-Mattson theorem [3,4] (see [5; Sec. 5.4]).
• \([23, 12, 7]\) Golay code— The dual of the Golay code is its \([23,11,8]\) even-weight subcode [6,7].
• \([n,n-1,2]\) Single parity-check (SPC) code— Binary SPCs and repetition codes are dual to each other.
• Hergert code— Hergert codes are duals of DG codes in that their distance distribution is equal to the MacWilliams transform of the distance distribution of DG codes [8]. However, the two codes are images of a pair of mutually dual linear codes over \(\mathbb{Z}_4\) under the Gray map [9,10].
• \([2^m-1,m,2^{m-1}]\) simplex code— Hamming and simplex codes are dual to each other.
• \([2^m,2^m-m-1,4]\) Extended Hamming code— Extended Hamming and first-order RM codes are dual to each other.
• \([7,4,3]\) Hamming code— The \([7,3,4]\) simplex code is the dual of the Hamming code and also its even-weight subcode [6,7].
• Reed-Muller (RM) code— The codes RM\((r,m)\) and RM\((m-r-1,m)\) are dual to each other, with the case \(m = 2r+1\) being self dual.
• Low-density generator-matrix (LDGM) code— The dual of an LDPC code has a sparse generator matrix and is called an LDGM code.
• Low-density parity-check (LDPC) code— The dual of an LDPC code has a sparse generator matrix and is called an LDGM code.
• Hsu-Anastasopoulos LDPC (HA-LDPC) code— HA-LDPC and MN-LDPC codes are dual to each other [11].
• Generalized RS (GRS) code— The dual of a GRS code is another GRS code [1; pg. 304].
• Reed-Solomon (RS) code— The dual of an RS code is an RS code [1; pg. 296].
• Generalized RM (GRM) code— The dual of an affine GRM code is also an affine GRM code [12,13].
• Maximum distance separable (MDS) 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 [14; Thm. 1.9.13].
• Dual additive code— Different inner products are typically used to define duals of linear and additive codes.
• \(q\)-ary simplex code— \(q\)-ary Hamming and \(q\)-ary simplex codes are dual to each other [15; pg. 45]
• \(q\)-ary LDGM code— The dual of a \(q\)-ary LDPC code has a sparse generator matrix and is called a \(q\)-ary LDGM code.
• \(q\)-ary LDPC code— The dual of a \(q\)-ary LDPC code has a sparse generator matrix and is called a \(q\)-ary LDGM code.
• Majorana stabilizer code— Classical self-orthogonal codes can be used to construct Majorana stabilizer codes [16–18]. The direct relationship between the two codes follows from expressing the Majorana strings as binary vectors – akin to the symplectic representation – and observing that the binary stabilizer matrix \(S\) for such a Majorana stabilizer code satisfies \(S\cdot S^T=0\) because it has commuting stabilizers, which is precisely the condition \(G\cdot G^T=0\) on the generator matrix \(G\) of a self-orthogonal classical code. A self-orthogonal classical code \(C\) with parameters \([2N,k,d]\) yields a Majorana stabilizer code with parameters \([[N,N-k,d_{\rm f}]]_f\), where \(d_{\rm f}=\min_{v\in C^\perp\setminus C}\mathrm{wt}(v)\geq d^\perp\) and \(d^\perp\) is the code distance of the dual code \(C^\perp\) [19].
• Permutationally self-dual (PSD) CSS code— The \(X\)- and \(Z\)-type stabilizer spaces of a permutationally self-dual CSS code are permutation-equivalent binary linear codes.
• Self-dual CSS code— Self-dual CSS codes arise from dual-containing (equivalently, self-orthogonal a.k.a. weakly self-dual) binary linear codes.
• Hermitian qubit code— Hermitian qubit codes are constructed from Hermitian self-orthogonal linear codes over \(\mathbb{F}_4\) via the \(\mathbb{F}_4\) representation.
• Qubit stabilizer code— Qubit stabilizer codes are in one-to-one correspondence with symplectic self-orthogonal binary linear codes of length \(2n\) via the symplectic representation.
• Hermitian Galois-qudit code— Hermitian codes are constructed from Hermitian self-orthogonal linear codes over \(\mathbb{F}_{q^2}\) via the \(\mathbb{F}_{q^2}\) representation.