Description
Reed-Muller code GRM\(_q(r,m)\) of length \(n=q^m\) over \(GF(q)\) with \(0\leq r\leq m(q-1)\). Its codewords are evaluations of the set of all degree-\(\leq r\) polynomials in \(m\) variables at the points of \(GF(q)\).
Since \(\beta^q=\beta\) for any \(\beta\in GF(q)\), the above definition is not injective. Replacing each factor in each polynomial as \(x^q\to x\), the above set reduces to the set of all degree-\(\leq r\) polynomials in \(m\) variables such that no term has an exponent \(q\) or higher on any variable.
Its automorphism group is the general affine group \(GA(m,GF(q))\) [4]. Any nontrivial \(q\)-ary linear code invariant under this group is equivalent to a GRM code [5].
Protection
Code parameters for specific \(m,r\) are given in Ref. [6][7; pg. 46].
Rate
GRM codes achieve capacity on sufficiently symmetric non-binary channels [8].
Notes
Parents
- Polynomial evaluation code — GRM (PRM) codes are multivariate polynomial evaluation codes with \(\cal X\) being the entire \(m\)-dimensional affine (projective) space over \(GF(q)\) [12,13][7; pgs. 44-46].
- Multiplicity code — Multivariate multiplicity codes of degree \(s=1\) reduce to GRM codes.
- Matrix-product code — Applying a special case of the matrix-product procedure yields GRM codes [14].
- \(q\)-ary linear LCC — GRM codes are LDCs and LCCs [15,16].
Children
- Reed-Muller (RM) code — Binary GRM codes are RM codes.
- Extended GRS code — GRM codes for univariate polynomials (\(m=1\)) reduce to extended RS codes [17].
- Projective RM (PRM) code
Cousins
- Group-algebra code — GRM codes over prime-power fields are group-algebra codes [18–20][21; Ex. 16.4.11].
- Cyclic linear \(q\)-ary code — GRM codes with nonzero evaluation points are cyclic [7; pg. 52].
- \(q\)-ary linear LTC — GRM codes for \(r<q\) can be LTCs in the low- [22,23] and high-error [24,25] regimes. They admit weakly stable presentations of their corresponding groups [26].
- Difference-set cyclic (DSC) code — DSC codes can be subfield subcodes of GRM codes, and visa versa [27; Thm. 6.14].
- Finite-geometry LDPC (FG-LDPC) code — Some EG-LDPC codes are duals of subfield subcodes of GRM codes [28; pg. 448].
- Batch code — GRM codes can be used to construct batch codes [29].
- \(q\)-ary Hamming code — \(q\)-ary Hamming codes are dual to first-order GRM codes [7; pg. 45].
- Galois-qudit quantum RM code
- Galois-qudit expander code — Expander codes with RS inner codes contain GRM codewords because tensor products of univariate polynomials (corresponding to RS codewords) yield multivariate polynomials (corresponding to GRM codewords) [30].
References
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Page edit log
- Victor V. Albert (2022-07-20) — most recent
Cite as:
“Generalized RM (GRM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_reed_muller