Here is a list of nonlinear codes related to linear codes via the Gray map.

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Code Description Relation
Delsarte-Goethals (DG) code Member of a family of \((2^{2t+2},2^{(2t+1)(t-d+2)+2t+3},2^{2t+1}-2^{2t+1-d})\) binary nonlinear codes for parameters \(r \geq 1\) and \(m = 2t+2 \geq 4\), denoted by DG\((m,r)\), that generalizes the Kerdock code. DG codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\) [1,2].
Gray code The first Gray code [3], now called the binary reflected Gray code, is a trivial \([n,n,1]\) code that orders length-\(n\) binary strings such that nearest-neighbor strings differ by only one digit via what is known as the Gray map.
Hergert code A nonlinear subcode of an RM code that is a formal dual of the nonlinear DG code in the sense that its distance distribution is equal to the MacWilliams transform of the distance distribution of a DG codes. Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [1,2,4].
Julin-Golay code One of several nonlinear binary \((12,144,4)\) codes based on the Steiner system \(S(5,6,12)\) [5,6][7; Sec. 2.7][8; Sec. 4] or their shortened versions, the nonlinear \((11,72,4)\), \((10,38,4)\), and \((9,20,4)\) Julin-Golay codes. Several of these codes contain more codewords than linear codes of the same length and distance and yield non-lattice sphere-packings that hold records in 9 and 11 dimensions. Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [8].
Kerdock code Binary nonlinear \((2^m, 2^{2m}, 2^{m-1} - 2^{(m-2)/2})\) for even \(m\) consisting of the first-order Reed-Muller code RM\((1,m)\) with maximum-rank cosets of RM\((1,m)\) in RM\((2,m)\). The image of the Kerdock code under the Gray map is the QRM\((1,m)\) code, an extended cyclic code over \(\mathbb{Z}_4\) [2; Thm. 19] (see also Ref. [9]).
Linear code over \(\mathbb{Z}_4\) A code that forms a subgroup of \(\mathbb{Z}_4^n\) under addition. More technically, linear codes over \(\mathbb{Z}_4\) are submodules of \(\mathbb{Z}_4^n\). A linear code \(C\) over \(\mathbb{Z}_4\) can be mapped, via the Gray map, to a binary code. The binary code is linear if and only if doubling the component-wise product of any two codewords in \(C\) yields another codeword in \(C\) [10; Thm. 12.2.3].
Phase-shift keying (PSK) code A \(q\)-ary phase-shift keying (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\). 1D Gray codes are often concatenated with PSKs so that the Hamming distance between the bitstrings encoded into the points is a discretized version of the Euclidean distance between the points.
Preparata code A nonlinear binary \((2^{m+1}-1, 2^{m+1}-2m-2, 5)\) code where \(m\) is odd. The size of this code is twice the size of the largest possible linear code with the same length and distance. The image of the Preparata code under the Gray map is the QRM\((m-2,m)\) code [2; Thm. 19].
Quadrature-amplitude modulation (QAM) code Encodes into points into a subset of points lying on in \(\mathbb{R}^{2}\), here treated as \(\mathbb{C}\). Each pair of points is associated with a complex amplitude of an electromagnetic signal, and information is encoded into both the norm and phase of that signal [11; Ch. 16]. 2D Gray codes are often concatenated with \(n=1\) lattice-based QAM codes so that the Hamming distance between the bitstrings encoded into the points is a discretized version of the Euclidean distance between the points.
Quaternary RM (QRM) code A quaternary linear code over \(\mathbb{Z}_4\) whose mod-two reduction is an RM code. This code subsumes the quaternary images of the Kerdock and Preparata codes under the Gray map . The code is usually noted as QRM\((r,m)\), with its mod-two reduction yielding the RM code RM\((r,m)\) [2; Thm. 19]. The mod-two reduction of the QRM\((r,m)\) code is the RM\((r,m)\) code [2; Thm. 19].
Qubit code Encodes \(K\)-dimensional Hilbert space into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space. Usually denoted as \(((n,K))\) or \(((n,K,d))\), where \(d\) is the code's distance. Gray codes are useful for optimizing qubit unitary circuits [12] and for encoding qudits in multiple qubits [13].
Rank-modulation code A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes [14], quadratic-residue codes, and most binary codes. The rank-modulation Gray code is an extension of the original binary Gray code to a code on the permutation group [15].
ZRM code A quaternary linear code over \(\mathbb{Z}_4\) that reduces to the RM code under the Gray map. The code is usually denoted as ZRM\((r,m-1)\), with its image under the Gray map being the RM code RM\((r,m)\) [2; Thm. 7]. The code is generated by \(\textnormal{RM}(r-1,m-1) + 2\textnormal{RM}(r,m-1)\) [2; Thm. 7]. The image of the ZRM\((r,m-1)\) code under the Gray map is the RM\((r,m)\) code [2; Thm. 7].
\(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code Member of a family of \(((2^m,2^{2^m−5m+1},8))\) CSS-like union stabilizer codes constructed using the classical Goethals and Preparata codes. A construction using the \(\mathbb{Z}_4\) versions of the Goethals and Preparata codes and the Gray map yields qubit code families with similar parameters as the \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code [16].
\((10,40,4)\) Best code Binary nonlinear \((10,40,4)\) code that is unique [17]. Under Construction A, this code yields \(P_{10c}\), a non-lattice sphere packing that is the densest known in 10 dimensions [18][19; pg. 140]. Codewords of the Best code can be obtained by applying the Gray map to the pentacode [8; Sec. 2].

References

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Anxiao Jiang, R. Mateescu, M. Schwartz, and J. Bruck, “Rank Modulation for Flash Memories”, IEEE Transactions on Information Theory 55, 2659 (2009) DOI
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