Generalized RM (GRM) code

Generalized RM (GRM) code[1][2][3]

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.

Notes

See books [5][6][7] for details of GRM codes.

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; [8][9]).
Matrix-product code — Applying a special case of the matrix-product procedure yields GRM codes [10].

Children

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

Cousins

Reed-Muller (RM) code
Cyclic linear \(q\)-ary code — GRM codes with nonzero evaluation points are cyclic ([4], pg. 52).
Extended GRS code — GRM codes for univariate polynomials (\(m=1\)) reduce to extended RS codes [11].
\(q\)-ary linear LTC — GRM codes for \(r<q\) can be LTCs in the low- [12][13] and high-error [14][15] regimes.
Group-algebra code — GRM codes over prime-power fields are group-algebra codes [16][17][18][19; Ex. 16.4.11].
\(q\)-ary Hamming code — Hamming codes are dual to first-order GRM codes ([4], pg. 45).

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]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[5]: E. F. Assmus and J. D. Key, Designs and Their Codes (Cambridge University Press, 1992) DOI
[6]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[7]: 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.
[8]: S. G. Vléduts and Yu. I. Manin, “Linear codes and modular curves”, Journal of Soviet Mathematics 30, 2611 (1985) DOI
[9]: T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[10]: T. Blackmore and G. H. Norton, “Matrix-Product Codes over ? q”, Applicable Algebra in Engineering, Communication and Computing 12, 477 (2001) DOI
[11]: J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
[12]: 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
[13]: S. Arora et al., “Proof verification and the hardness of approximation problems”, Journal of the ACM 45, 501 (1998) DOI
[14]: 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
[15]: 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
[16]: S. D. Berman, “On the theory of group codes”, Cybernetics 3, 25 (1969) DOI
[17]: Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.
[18]: P. Landrock and O. Manz, “Classical codes as ideals in group algebras”, Designs, Codes and Cryptography 2, 273 (1992) DOI
[19]: W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/ag/generalized_reed_muller.yml.