## 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

## Rate

## 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

## References

- [1]
- T. Kasami, Shu Lin, and W. Peterson, “New generalizations of the Reed-Muller codes--I: Primitive codes”, IEEE Transactions on Information Theory 14, 189 (1968) DOI
- [2]
- E. Weldon, “New generalizations of the Reed-Muller codes--II: Nonprimitive codes”, IEEE Transactions on Information Theory 14, 199 (1968) DOI
- [3]
- P. Delsarte, J. M. Goethals, and F. J. Mac Williams, “On generalized ReedMuller codes and their relatives”, Information and Control 16, 403 (1970) DOI
- [4]
- T. Berger and P. Charpin, “The automorphism group of Generalized Reed-Muller codes”, Discrete Mathematics 117, 1 (1993) DOI
- [5]
- P. Delsarte, “On cyclic codes that are invariant under the general linear group”, IEEE Transactions on Information Theory 16, 760 (1970) DOI
- [6]
- M. Tsfasman, S. Vlǎduţ, and D. Nogin. Algebraic geometric codes: basic notions. Vol. 139. American Mathematical Society, 2022.
- [7]
- M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
- [8]
- G. Reeves and H. D. Pfister, “Achieving Capacity on Non-Binary Channels with Generalized Reed-Muller Codes”, (2023) arXiv:2305.07779
- [9]
- E. F. Assmus and J. D. Key, Designs and Their Codes (Cambridge University Press, 1992) DOI
- [10]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [11]
- E. F. Assmus, Jr. and J. D. Key, “Polynomial codes and finite geometries,” in Handbook of Coding Theory, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp. 1269–1343.
- [12]
- S. G. Vléduts and Yu. I. Manin, “Linear codes and modular curves”, Journal of Soviet Mathematics 30, 2611 (1985) DOI
- [13]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
- [14]
- T. Blackmore and G. H. Norton, “Matrix-Product Codes over ? q”, Applicable Algebra in Engineering, Communication and Computing 12, 477 (2001) DOI
- [15]
- S. Yekhanin, “Locally Decodable Codes”, Foundations and Trends® in Theoretical Computer Science 6, 139 (2012) DOI
- [16]
- Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.
- [17]
- J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
- [18]
- S. D. Berman, “On the theory of group codes”, Cybernetics 3, 25 (1969) DOI
- [19]
- Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.
- [20]
- P. Landrock and O. Manz, “Classical codes as ideals in group algebras”, Designs, Codes and Cryptography 2, 273 (1992) DOI
- [21]
- W. Willems, "Codes in Group Algebras." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [22]
- L. Babai et al., “Checking computations in polylogarithmic time”, Proceedings of the twenty-third annual ACM symposium on Theory of computing - STOC ’91 (1991) DOI
- [23]
- S. Arora et al., “Proof verification and the hardness of approximation problems”, Journal of the ACM 45, 501 (1998) DOI
- [24]
- R. Raz and S. Safra, “A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP”, Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC ’97 (1997) DOI
- [25]
- S. Arora and M. Sudan, “Improved low-degree testing and its applications”, Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC ’97 (1997) DOI
- [26]
- M. Chapman, T. Vidick, and H. Yuen, “Efficiently stable presentations from error-correcting codes”, (2023) arXiv:2311.04681
- [27]
- C. Ding, Codes from Difference Sets (WORLD SCIENTIFIC, 2014) DOI
- [28]
- R. E. Blahut, Algebraic Codes for Data Transmission (Cambridge University Press, 2003) DOI
- [29]
- T. Baumbaugh et al., “Batch Codes from Hamming and Reed-Müller Codes”, (2017) arXiv:1710.07386

## 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