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

Child

Cousins

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 Y. 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]
John B. Little, “Algebraic geometry codes from higher dimensional varieties”. 0802.2349

Zoo code information

Internal code ID: generalized_reed_muller

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Zoo Code ID: generalized_reed_muller

Cite as:
“Generalized RM (GRM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_reed_muller
BibTeX:
@incollection{eczoo_generalized_reed_muller, title={Generalized RM (GRM) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/generalized_reed_muller} }
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“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/eval/generalized_reed_muller.yml.