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
Page edit log
- Victor V. Albert (2022-08-10) — most recent
Cite as:
“Projective RM (PRM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/projective_reed_muller