Alternative names: \(\mathbb{R}P^N\) code, Packing in \(\mathbb{R}P^N\).
Description
Encodes \(K\) states (codewords) into a real projective space \(\mathbb{R}P^N\), the space of lines in real space. The space for \(N=2\) is called the projective plane.Protection
A common distance measure on real projective space is the inner product (a.k.a. coherence). Sets of lines with minimal inner product are called Grassmanian frames [1], and most of these frames are equiangular tight frames [2]. The Welch bound is a lower bound on the worst-case coherence of a code [3].Cousins
- Constant-energy spherical code— Real projective space can be obtained from the sphere by identifying antipodal points, i.e., \(\mathbb{R}P^N = S^N/\mathbb{Z}_2\). As such, real projective space codes are in one-to-one correspondence with antipodal spherical codes, with each antipodal pair of spherical codewords corresponding to one line in projective space.
- Kerdock code— The 12 sets of antipodal pairs of the 24-cell code form a sharp configuration in the projective space \(\mathbb{R}P^3\) [4]. This is a special case of a family of real projective plane codes, constructed using Kerdock codes [5] (cf. [6]).
- \(E_7\) lattice-shell code— The 63 sets of antipodal pairs of the smallest \(E_7\) lattice shell form a sharp configuration in the projective space \(\mathbb{R}P^6\) [4].
- \(E_6\) lattice-shell code— The 36 sets of antipodal pairs of the smallest \(E_6\) lattice shell form a sharp configuration in the projective space \(\mathbb{R}P^5\) [4].
- Polygon code— The \(q/2\) sets of antipodal pairs of a \(q\)-gon form a tight design on the projective plane [6].
- 24-cell code— The 12 sets of antipodal pairs of the 24-cell code form a sharp configuration in the projective space \(\mathbb{R}P^3\) [4]. This is a special case of a family of real projective plane codes, constructed using Kerdock codes [5] (cf. [6]).
- Witting polytope code— Antipodal pairs of points of the Witting polytope code form a 3-design in \(\mathbb{R}P^7\) [4].
Member of code lists
Primary Hierarchy
Parents
Real projective spaces \(\mathbb{R}P^N\) are real Grassmannians \(G/H\) for \(G = O(N+1)\) and \(H = O(N)\times O(1)\).
Real projective space code
References
- [1]
- T. Strohmer and R. Heath, “Grassmannian Frames with Applications to Coding and Communication”, (2003) arXiv:math/0301135
- [2]
- M. Fickus and D. G. Mixon, “Tables of the existence of equiangular tight frames”, (2016) arXiv:1504.00253
- [3]
- L. Welch, “Lower bounds on the maximum cross correlation of signals (Corresp.)”, IEEE Transactions on Information Theory 20, 397 (1974) DOI
- [4]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [5]
- Levenshtein, V. I. (1982). Bounds on the maximal cardinality of a code with bounded modulus of the inner product. In Soviet Math. Dokl (Vol. 25, No. 2, pp. 526-531).
- [6]
- H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
Page edit log
- Victor V. Albert (2025-10-27) — most recent
Cite as:
“Real projective space code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/real_projective