\(k\)-orthogonal code[1–3]

Cousins

• Modular-qudit lattice color code— The notion of \(k\)-orthogonality can be extended to modular-qudit codes and is known as \(k^{\star}\)-orthogonality [1; Def. 2]. Modular-qudit lattice color codes defined on lattices in \(D\) spatial dimension whose \(X\)-type stabilizers are placed on cells of dimension \(\nu \leq D\) are \(k^{\star}\)-orthogonal for all \(k \leq \nu\) [1; Lemma 5].
• Reed-Muller (RM) code— A unital \(k\)-orthogonal space is a \(k\)-orthogonal binary space containing the all-ones vector. For dimension \(m+1\), its generator matrix has columns \((1,y)\) for \(y\) in a subset \(Y\subseteq\mathbb{F}_2^m\), and the overlap of any \(j\leq k\) rows, repetitions allowed, is the sum over \(Y\) of a monomial of degree at most \(j\). The indicator function of \(Y\) is therefore orthogonal to \(RM(k,m)\), i.e., it is a codeword of \(RM(m-k-1,m)\). For \(k=3\), unital triorthogonal spaces correspond one-to-one to affine-equivalence classes of \(RM(m-4,m)\) codewords whose weight equals the length of the space and that are not divisible by a nonconstant affine function [5][6; Supp. Sec. III]. The correspondence yielded a classification of unital triorthogonal spaces through length 38 via the Kasami-Tokura classification of low-weight RM codewords [5]. It was extended through length 54 by a directional-derivative recursion that reconstructs a codeword from its derivative along a direction with the fewest pairs of support points [6; Supp. Sec. III].
• \([[2^r-1,1,3]]\) simplex code— \([[2^r-1,1,3]]\) simplex codes are \((r-1)\)-orthogonal [3; Lemma 2].