## 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 a set of distinct points \(\{\alpha_1,\cdots,\alpha_n\}\) in \(GF(q)\).

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.

## Protection

Code parameters for specific \(m,r\) are given in Ref. [4], pg. 46.

## Rate

GRM codes achieve capacity on sufficiently symmetric non-binary channels [5].

## 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 \(GF(q)\) ([4], pgs. 44-46; [9,10]).
- 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 [11].
- \(q\)-ary linear LCC — GRM codes are LDCs and LCCs [12,13].
- Group-algebra code — GRM codes over prime-power fields are group-algebra codes [14–16][17; Ex. 16.4.11].

## Children

- Reed-Muller (RM) code — Binary GRM codes are RM codes.
- Projective RM (PRM) code

## Cousins

- Cyclic linear \(q\)-ary code — GRM codes with nonzero evaluation points are cyclic ([4], pg. 52).
- \(q\)-ary linear LTC — GRM codes for \(r<q\) can be LTCs in the low- [18,19] and high-error [20,21] regimes. They admit weakly stable presentations of their correspoding groups [22].
- Finite-geometry LDPC (FG-LDPC) code — Some EG-LDPC codes are duals of subfield subcodes of GRM codes [23; pg. 448].
- Extended GRS code — GRM codes for univariate polynomials (\(m=1\)) reduce to extended RS codes [24].
- \(q\)-ary Hamming code — \(q\)-ary Hamming codes are dual to first-order GRM codes [4; pg. 45].
- Galois-qudit quantum RM code

## 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