Projective RM (PRM) code[1,2] 


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}



  • 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].


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

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“Projective RM (PRM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@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={} }
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“Projective RM (PRM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.