Reed-Muller (RM) code[1–3] 

Description

Member of the RM\((r,m)\) family of linear binary codes derived from multivariate polynomials. The code parameters are \([2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]\), where \(r\) is the order of the code satisfying \(0\leq r\leq m\). First-order RM codes are also called biorthogonal codes, while \(m\)th order RM codes are also called universe codes. Punctured RM codes RM\(^*(r,m)\) are obtained from RM codes by deleting one coordinate from each codeword.

Generator matrices of RM codes are constructed using the \((u|u+v)\) construction by starting from the \(2^m\)-dimensional matrix \(F^{(m)}=\left(\begin{smallmatrix}1 & 0\\ 1 & 1 \end{smallmatrix}\right)^{\otimes m}\), labeling its rows top-to-bottom from \(0\) to \(2^m-1\), converting the labels to binary strings of length \(m\), and deleting all rows whose labels have a Hamming weight less than \(m-r\). The recursive nature of the tensor product in the matrix \(F^{(m)}\) implies that RM\((r,m)\) is a subcode of RM\((r+1,m)\).

Another way to generate RM codewords is to list all outcomes of all polynomials of \(m\) binary variables of degree at most \(r\) [4] (see also Ch. 13 of Ref. [5]).

The automorphism code of the RM\((r,m)\) (RM\(^*(r,m)\)) code is \(GA_{m}(\mathbb{F}_2)\) (\(GL_{m}(\mathbb{F}_2)\)) for \(1 \leq r \leq m-2\) [5].

Protection

The Schwartz-Zippel Lemma provides a distance lower bound on RM codes and extended the degree mantra from RS codes.

Rate

Achieves capacity of the binary erasure channel [6,7], the binary memoryless symmetric (BMS) channel under bitwise maximum-a-posteriori decoding [8] (see also Ref. [9]), and the binary symmetric channel (BSC), solving a long-standing conjecture [10].

Decoding

Reed decoder with \(r+1\)-step majority decoding corrects \(\frac{1}{2}(2^{m-r}-1)\) errors [1] (see also Ch. 13 of Ref. [5]).Sequential code-reduction decoding [11].Matrix factorization is equivalent to decoding an RM\((n,n-3)\) code [12]; see [13].

Realizations

Deep-space communication [14,15].

Notes

See [16][5; Chs. 13-15] for details of RM codes and their variants.

Parents

  • Linear binary code
  • \((u|u+v)\)-construction code — All RM codes can be constructed via the \((u|u+v)\) construction [5; Ch. 13].
  • Generalized RM (GRM) code — Binary GRM codes are RM codes.
  • Divisible code — An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece's theorem [17,18].
  • Group-algebra code — RM codes are group-algebra codes [19,20][21; Ex. 16.4.11]. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary Abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the group algebra \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).

Children

Cousins

  • Binary BCH code — RM\(^*(r,m)\) codes are equivalent to subcodes of BCH codes of designed distance \(2^{m-r}-1\) while RM\((r,m)\) are subcodes of extended BCH codes of the same designed distance [5; Ch. 13].
  • Quaternary linear code over \(\mathbb{Z}_4\) — RM codes are images of linear quaternary codes over \(\mathbb{Z}_4\) under the Gray map [22; Sec. 6.3].
  • Gray code — RM codes are images of linear quaternary codes over \(\mathbb{Z}_4\) under the Gray map [22; Sec. 6.3].
  • Dual linear code — The codes RM\((r,m)\) and RM\((m-r-1,m)\) are dual to each other.
  • Binary duadic code — Certain punctured RM codes such as RM\(^*(2,5)\) are duadic; see Ref. [23], Table 6.2.
  • Cyclic linear binary code — Punctured RM codes are cyclic ([5], Ch. 13, Thm. 11), making RM codes extended cyclic codes. RM codes with nonzero evaluation points are cyclic [24][25; pg. 52].
  • Binary linear LTC — RM codes can be LTCs in the low- [26,27] and high-error [28] regimes; see also [29].
  • Qubit code — Optimizing T gates in a qubit circuit that uses CNOT and T gates is equivalent decoding a paricular RM code [30].
  • Orthogonal array (OA) — RM codes are related to orthogonal arrays [31; Exam. 10.57].
  • Barnes-Wall (BW) lattice code — BW lattice codes are lattice analogues of RM codes in that both can be constructed recursively via a \(|u|u+v|\) construction [32,33].
  • \(\Lambda_{16}\) Barnes-Wall lattice code — The RM\((1,4)\) code can be used to obtain the \(\Lambda_{16}\) Barnes-Wall lattice code [34; Ex. 10.7.2].
  • Combinatorial design — Fixed-weight RM codewords of weight less than \(2^m\) support combinatorial 3-designs [35; Ex. 5.2.7].
  • Hadamard code — The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)). The \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)).
  • \([2^r-1,2^r-r-1,3]\) Hamming code — Binary Hamming codes are equivalent to RM\(^*(r-2,r)\).
  • Preparata code — A Preparata code can be written as a union of a linear subcode \(\mathcal{C}\) of RM\((m-2,m)\) and the \(2^{m-1}-1\) representatives of coset formed by \(\mathcal{C}\) in RM\((m-2,m)\). The coset representatives are given by \(|1|x^j|0|x^{j}\theta_{1}|\), where \(1\leq j \leq 2^{m-1}-1\). \(\mathcal{C}\) comprises of codewords of the form \(|g(1)|g(x)(1+\theta_{1})|f(1)+g(1)|g(x)(1+\theta_{1})+f(x)(1+\theta_{1}+\theta_{3})|\), where \(f(x)\) and \(g(x)\) are arbitrary, and where \(\theta_{1}\) and \(\theta_{3}\) denote the primitive idempotents corresponding to cyclotomic cosets \(C_1\) and \(C_3\) respectively.
  • Delsarte-Goethals (DG) code — The code DG\((m,r)\) is a subcode of the second-order Reed-Muller code RM\((2,m)\), and equals RM\((2,m)\) at \(r=1\) [5; pg. 461]. The code is the union of certain cosets of the first-order RM\((1,m)\) code in RM\((2,m)\) that are specified by bilinear forms [36].
  • Kerdock code — Kerdock code is a subcode of a second-order RM Code [5; pg. 457]. It consists of a number of cosets of RM\((2,m)\) created by quotienting with first-order RM\((1,m)\) codes.
  • Polar code — The generator matrices of RM and polar codes are different submatrices of Kronecker products of Hadamard matrices; see Ref. [37]. There are families interpolating between the two codes [38].
  • Majorana stabilizer code — Majorana stabilizer codes can be constructed by self-orthogonal RM codes [39]. These codes have the additional property that the global fermion parity is fixed in the codespace. In this family of codes, logical measurements are reduced to parity measurements of some subset of Majorana fermions in the code.
  • Quantum Reed-Muller code

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“Reed-Muller (RM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/reed_muller
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@incollection{eczoo_reed_muller, title={Reed-Muller (RM) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/reed_muller} }
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