Projective RM (PRM) code[1,2] 

Description

Reed-Muller code for nonzero points \(\{\alpha_1,\cdots,\alpha_n\}\) whose leftmost nonzero coordinate is one, corresponding to an evaluation code of polynomials over projective coordinates.

PRM codes PRM\(_q(r,m)\) for \(r<q\) are injective evaluation codes with parameters [3] \begin{align} \left[ q^m+q^{m-1}\cdots +1, {m+r \choose r},(q+1-r)q^{m-1} \right]~. \tag*{(1)}\end{align}

Parent

Cousins

  • Projective geometry code — Nonzero codewords of minimum weight of a \(r\)th-order \(q\)-ary projective RM code correspond to algebraic hypersurfaces of degree \(r\) having the largest number of points in the projective space \(PG(n,q)\) [4; Thm. 14.3.3].
  • Griesmer code — PRM codes for \(r=1\) attain the Griesmer bound for all \(m\) [5].

References

[1]
G. Lachaud, “The parameters of projective Reed–Müller codes”, Discrete Mathematics 81, 217 (1990) DOI
[2]
A. B. Sorensen, “Projective Reed-Muller codes”, IEEE Transactions on Information Theory 37, 1567 (1991) DOI
[3]
G. Lachaud, “Number of points of plane sections and linear codes defined on algebraic varieties”, Arithmetic, Geometry, and Coding Theory DOI
[4]
L. Storme, "Coding Theory and Galois Geometries." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[5]
J. B. Little, “Algebraic geometry codes from higher dimensional varieties”, (2008) arXiv:0802.2349
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Zoo Code ID: projective_reed_muller

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

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