From 43f8e8e58eb83a975cea89ca5a31f866a5e16d88 Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Tue, 23 Jul 2024 19:44:01 -0400 Subject: [PATCH] ~ --- codes/classical/bits/easy/dual_hamming/hadamard.yml | 11 +---------- .../classical/bits/easy/hamming/extended_hamming.yml | 6 +++++- codes/classical/bits/easy/hamming/hamming.yml | 3 ++- codes/classical/bits/polar.yml | 2 +- codes/classical/bits/reed_muller/biorthogonal.yml | 4 +++- codes/classical/bits/reed_muller/reed_muller.yml | 4 ++-- codes/classical/q-ary_digits/ag/ag.yml | 2 +- .../q-ary_digits/ag/evaluationAG/klein_quartic.yml | 4 ++-- .../q-ary_digits/ag/generalized_reed_muller.yml | 4 ++-- .../q-ary_digits/ag/projective_reed_muller.yml | 2 +- .../classical/q-ary_digits/ag/residueAG/residue.yml | 5 ++--- .../classical/q-ary_digits/ag/residueAG/shimura.yml | 4 +++- codes/classical/q-ary_digits/ag/rs/reed_solomon.yml | 12 ++++-------- 13 files changed, 29 insertions(+), 34 deletions(-) diff --git a/codes/classical/bits/easy/dual_hamming/hadamard.yml b/codes/classical/bits/easy/dual_hamming/hadamard.yml index c8e7a05cf..b46490a07 100644 --- a/codes/classical/bits/easy/dual_hamming/hadamard.yml +++ b/codes/classical/bits/easy/dual_hamming/hadamard.yml @@ -15,7 +15,7 @@ alternative_names: - 'Walsh-Hadamard code' description: | - An \([2^m,m,2^{m-1}]\) balanced binary code dual to an extended Hamming code. + An \([2^m,m,2^{m-1}]\) balanced binary code. The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)), while the \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)). @@ -33,15 +33,6 @@ relations: cousins: - code_id: long detail: 'The Hadamard code is a subcode of the long code and can be obtained by restricting the long-code construction to only linear functions.' - - code_id: dual - detail: 'The Hadamard code is the dual of the extended Hamming Code. - Conversely, the shortened Hadamard code is the dual of the Hamming Code.' - - code_id: hamming - detail: 'The Hadamard code is the dual of the extended Hamming Code. - Conversely, the shortened Hadamard code is the dual of the Hamming Code.' - - code_id: extended_hamming - detail: 'The Hadamard code is the dual of the extended Hamming Code. - Conversely, the shortened Hadamard code is the dual of the Hamming Code.' - code_id: reed_muller detail: 'The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)), while the \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)).' - code_id: delsarte_optimal_q-ary diff --git a/codes/classical/bits/easy/hamming/extended_hamming.yml b/codes/classical/bits/easy/hamming/extended_hamming.yml index ba2996fb1..ba933525c 100644 --- a/codes/classical/bits/easy/hamming/extended_hamming.yml +++ b/codes/classical/bits/easy/hamming/extended_hamming.yml @@ -7,7 +7,7 @@ code_id: extended_hamming physical: bits logical: bits -name: 'Extended Hamming code' +name: '\([2^r,2^r-r-1,4]\) Extended Hamming code' introduced: '\cite{doi:10.1002/j.1538-7305.1948.tb01338.x,doi:10.1002/j.1538-7305.1950.tb00463.x,manual:{M. J. E. Golay, \emph{Notes on digital coding}, Proc. IEEE, 37 (1949) 657.}}' description: | @@ -20,6 +20,10 @@ relations: cousins: - code_id: univ_opt_q-ary detail: 'Several extended Hamming codes are LP universally optimal codes \cite{arxiv:1212.1913}.' + - code_id: biorthogonal + detail: 'Extended Hamming and first-order RM codes are dual to each other.' + - code_id: dual + detail: 'Extended Hamming and first-order RM codes are dual to each other.' # Begin Entry Meta Information diff --git a/codes/classical/bits/easy/hamming/hamming.yml b/codes/classical/bits/easy/hamming/hamming.yml index bb156a497..931e6c94b 100644 --- a/codes/classical/bits/easy/hamming/hamming.yml +++ b/codes/classical/bits/easy/hamming/hamming.yml @@ -16,7 +16,7 @@ alternative_names: description: | Member of an infinite family of perfect linear codes with parameters \([2^r-1,2^r-r-1, 3]\) for \(r \geq 2\). Their \(r \times (2^r-1) \) parity-check matrix \(H\) has all possible non-zero \(r\)-bit strings as its columns. - Adding a parity check yields the \([2^r,2^r-r-1, 4]\) \textit{extended Hamming code}. + Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code. protection: 'Can detect 1-bit and 2-bit errors, and can correct 1-bit errors.' @@ -44,6 +44,7 @@ relations: - code_id: constantin_rao detail: 'The nonlinear CR codes for \(G = \mathbb{Z}_2^r\) reduce to Hamming codes at lengths \(n = 2^r - 1\) \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}; see Ref. \cite{arxiv:1310.7536}.' - code_id: extended_hamming + detail: 'Extended Hamming codes are extensions of Hamming codes by a parity-check bit.' - code_id: reed_muller detail: 'Binary Hamming codes are equivalent to RM\(^*(r-2,r)\).' - code_id: nearly_perfect diff --git a/codes/classical/bits/polar.yml b/codes/classical/bits/polar.yml index 43dfe35a7..7599d4bd9 100644 --- a/codes/classical/bits/polar.yml +++ b/codes/classical/bits/polar.yml @@ -48,7 +48,7 @@ relations: detail: 'Polar codes can be represented as generalized concatenations of their kernels.' cousins: - code_id: reed_muller - detail: 'RM codes rely on the same generator matrix, but place message bits in different coordinates; see Ref. \cite{doi:10.1109/ITWKSPS.2010.5503223}. There are families interpolating between the two codes \cite{doi:10.1109/TCOMM.2014.2345069}.' + detail: 'The generator matrices of RM and polar codes are different submatrices of Kronecker products of Hadamard matrices; see Ref. \cite{doi:10.1109/ITWKSPS.2010.5503223}. There are families interpolating between the two codes \cite{doi:10.1109/TCOMM.2014.2345069}.' - code_id: polar_for_quantum detail: 'Quantum-classical polar codes generalize polar codes for transmission through channels with quantum output.' diff --git a/codes/classical/bits/reed_muller/biorthogonal.yml b/codes/classical/bits/reed_muller/biorthogonal.yml index 2986c4507..31246a312 100644 --- a/codes/classical/bits/reed_muller/biorthogonal.yml +++ b/codes/classical/bits/reed_muller/biorthogonal.yml @@ -30,7 +30,9 @@ realizations: - 'The \([32, 6, 16]\) RM\((1,5)\) code was used for the 1971 Mariner 9 spacecraft \cite{doi:10.1109/ITWKSPS.2010.5503223}.' notes: - - 'See Ref. \cite{doi:10.1109/TIT.1972.1054732} for the weight distribution of the \(2^{26}\) cosets of the \([32,6]\) first-order RM code, obtained in part by hand computation.' + - 'See Ref. \cite{doi:10.1109/TIT.1972.1054732} for the weight distribution of the \(2^{26}\) cosets of the \([32,6]\) first-order RM code.' + +# , obtained in part by hand computation.' relations: diff --git a/codes/classical/bits/reed_muller/reed_muller.yml b/codes/classical/bits/reed_muller/reed_muller.yml index 9a14bf7d8..84603c672 100644 --- a/codes/classical/bits/reed_muller/reed_muller.yml +++ b/codes/classical/bits/reed_muller/reed_muller.yml @@ -14,7 +14,7 @@ introduced: '\cite{doi:10.1109/irepgelc.1954.6499441,doi:10.1109/tit.1954.105746 description: | Member of the RM\((r,m)\) family of linear binary codes derived from multivariate polynomials. The code parameters are \([2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]\), where \(r\) is the \textit{order} of the code satisfying \(0\leq r\leq m\). First-order RM codes are also called biorthogonal codes, while \(m\)th order RM codes are also called \textit{universe} codes. - \textit{Punctured RM codes} RM\(^*(r,m)\) are obtained from RM codes by deleting one or more coordinates from each codeword. + \textit{Punctured RM codes} RM\(^*(r,m)\) are obtained from RM codes by deleting one coordinate from each codeword. Generator matrices of RM codes are constructed using the \((u|u+v)\) construction by starting from the \(2^m\)-dimensional matrix \(F^{(m)}=\left(\begin{smallmatrix}1 & 0\\ 1 & 1 @@ -40,7 +40,7 @@ realizations: - 'Deep-space communication \cite{doi:10.1007/bfb0036046,manual:{E.C. Posner, \emph{Combinatorial Structures in Planetary Reconnaissance} in Error Correcting Codes, ed. H.B. Mann, Wiley, NY 1968.}}.' notes: - - 'See Chs. 13-15 of Ref. \cite{preset:MacSlo} for details of RM codes and their variants.' + - 'See \cite[Chs. 13-15]{preset:MacSlo}\cite{doi:10.1017/CBO9781316529836} for details of RM codes and their variants.' # If mobing to evaluation, then remove "nontrivial" in that entry relations: diff --git a/codes/classical/q-ary_digits/ag/ag.yml b/codes/classical/q-ary_digits/ag/ag.yml index 0334ad89f..6e3018ba4 100644 --- a/codes/classical/q-ary_digits/ag/ag.yml +++ b/codes/classical/q-ary_digits/ag/ag.yml @@ -18,7 +18,7 @@ description: | In alternative conventions (not used here), AG codes are restricted to be linear and/or include \hyperref[code:evaluation_varieties]{evaluation} codes defined using algebraic varieties more general than curves. features: - rate: 'Several sequences of linear AG codes beat the Gilbert-Varshamov bound and/or are asymptotically good \cite{doi:10.1007/BF01884295,doi:10.1006/jnth.1996.0147} (see Ref. \cite{preset:HPAlgCodes} for details). + rate: 'Several sequences of linear AG codes beat the Gilbert-Varshamov bound and/or are asymptotically good \cite{doi:10.1002/mana.19821090103,doi:10.1007/BF01884295,doi:10.1006/jnth.1996.0147} (see Ref. \cite{preset:HPAlgCodes} for details). The rate of any linear AG code satisfies \begin{align} \frac{k}{n} \geq 1 - \frac{d}{n} - \frac{1}{\sqrt{q}-1}~, diff --git a/codes/classical/q-ary_digits/ag/evaluationAG/klein_quartic.yml b/codes/classical/q-ary_digits/ag/evaluationAG/klein_quartic.yml index 9bc779774..ac86711f3 100644 --- a/codes/classical/q-ary_digits/ag/evaluationAG/klein_quartic.yml +++ b/codes/classical/q-ary_digits/ag/evaluationAG/klein_quartic.yml @@ -11,9 +11,9 @@ name: 'Klein-quartic code' introduced: '\cite{doi:10.1109/TIT.1987.1057365}' description: | - Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying in the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) (\cite{preset:HPAlgCodes}, Ex. 2.75). + Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) (\cite{preset:HPAlgCodes}, Ex. 2.75). -protection: 'Dimension \(k=8\) and distance \(d \geq 13\). Concatenation with the \([4,3,2]\) single parity check code, conversion to a binary code by expressing \(GF(8)\) elements as vectors over \(GF(2)\), and puncturing yields a \([91,24,25]\) binary code that held the world record for codes of length 91 \cite{manual:{A. M. Barg, G. L. Katsman, M. A. Tsfasman, “Algebraic-Geometric Codes from Curves of Small Genus”, Probl. Peredachi Inf., 23:1 (1987), 42–46; Problems Inform. Transmission, 23:1 (1987), 34–38}}.' +protection: 'Dimension \(k=8\) and distance \(d \geq 13\). Concatenation with the \([4,3,2]\) single parity check code, conversion to a binary code by expressing \(GF(8)\) elements as vectors over \(GF(2)\), and puncturing yields a \([91,24,25]\) binary code that set the world record for codes of length 91 \cite{manual:{A. M. Barg, G. L. Katsman, M. A. Tsfasman, “Algebraic-Geometric Codes from Curves of Small Genus”, Probl. Peredachi Inf., 23:1 (1987), 42–46; Problems Inform. Transmission, 23:1 (1987), 34–38}}.' relations: parents: diff --git a/codes/classical/q-ary_digits/ag/generalized_reed_muller.yml b/codes/classical/q-ary_digits/ag/generalized_reed_muller.yml index 93110cef2..1bf43f3b2 100644 --- a/codes/classical/q-ary_digits/ag/generalized_reed_muller.yml +++ b/codes/classical/q-ary_digits/ag/generalized_reed_muller.yml @@ -12,7 +12,7 @@ short_name: 'GRM' introduced: '\cite{doi:10.1109/TIT.1968.1054127,doi:10.1109/TIT.1968.1054128,doi:10.1016/S0019-9958(70)90214-7}' 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 \(GF(q)\). + 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 the points of \(GF(q)\). 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. @@ -38,9 +38,9 @@ relations: detail: 'Applying a special case of the matrix-product procedure yields GRM codes \cite{doi:10.1007/PL00004226}.' - code_id: q-ary_lcc detail: 'GRM codes are LDCs and LCCs \cite{doi:10.1561/0400000030,manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.' + cousins: - code_id: group detail: 'GRM codes over prime-power fields are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.},doi:10.1007/BF00141972}\cite[Ex. 16.4.11]{preset:HKSalgebra}.' - cousins: - code_id: q-ary_cyclic detail: 'GRM codes with nonzero evaluation points are cyclic \cite[pg. 52]{doi:10.1007/978-94-011-3810-9}.' - code_id: q-ary_ltc diff --git a/codes/classical/q-ary_digits/ag/projective_reed_muller.yml b/codes/classical/q-ary_digits/ag/projective_reed_muller.yml index 9a7c21204..1351b53f8 100644 --- a/codes/classical/q-ary_digits/ag/projective_reed_muller.yml +++ b/codes/classical/q-ary_digits/ag/projective_reed_muller.yml @@ -12,7 +12,7 @@ short_name: 'PRM' introduced: '\cite{doi:10.1016/0012-365X(90)90155-B,doi:10.1109/18.104317}' 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. + Reed-Muller code for nonzero points \(\{\alpha_1,\cdots,\alpha_n\}\) with \(n=m+1\) whose leftmost nonzero coordinate is one, corresponding to an evaluation code of polynomials over projective coordinates. PRM codes PRM\(_q(r,m)\) for \(r