~

codes/classical/q-ary_digits/ag/residueAG/goppa.yml

@@ -24,11 +24,12 @@ features:
   decoders:
     - 'Algebraic decoding algorithms \cite{doi:10.1109/TIT.1975.1055350}. If \( \text{deg} G(x) = 2t \) , then there exists a \(t\)-correcting algebraic decoding algorithm for \( \Gamma(L,G) \).'
     - 'Sugiyama et al. modification of the extended Euclidean algorithm \cite{doi:10.1016/S0019-9958(75)90090-X,doi:10.1017/CBO9780511606267}.'
-    - 'Guruswami-Sudan list decoder \cite{doi:10.1109/18.782097,doi:10.1109/SFCS.1998.743426}.'
     - 'Binary Goppa codes can be decoded using a RS-based decoder \cite{manual:{Daniel J. Bernstein, "Understanding binary-Goppa decoding." Cryptology ePrint Archive (2022).}}.'
+    - 'List decoder for binary Goppa codes \cite{doi:10.1109/TIT.2013.2243800}.'
 
 realizations:
-  - 'Initial version of the McEliece public-key cryptosystem \cite{manual:{R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Technical report, Jet Propulsion Lab. DSN Progress Report (1978).},doi:10.1007/BF00173300} and its variation by Niederreiter \cite{manual:{H. Niederreiter (1986). \emph{Knapsack-type cryptosystems and algebraic coding theory}. Problems of Control and Information Theory. Problemy Upravlenija I Teorii Informacii. 15: 159–166.}} where the generator matrix is replaced by the parity check matrix. Some of these were proven to be insecure since the public key exposes algebraic structure of code \cite{doi:10.1515/dma.1992.2.4.439}.'
+  - 'The McEliece public-key cryptosystem \cite{manual:{R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Technical report, Jet Propulsion Lab. DSN Progress Report (1978).},doi:10.1007/BF00173300}.'
+
 
 notes:
   - 'GAP function \href{https://www.gap-system.org/Manuals/pkg/guava/doc/chap5.html#X7EE808BB7D1E487A}{GoppaCode(G,L)} takes in a polynomial \(G\) that satisfies the necessary conditions for a Goppa code and a list \(L\) that contains elements in \(GF(q)\) that are not roots of \(G\). It returns a Goppa code.'

codes/classical/q-ary_digits/ag/rs/extended_reed_solomon.yml

@@ -14,6 +14,7 @@ description: |
 
   An \([q-1,k,q-k]_q\) narrow-sense RS code can be extended twice by adding two evaluation points (of which one can be zero) to yield a \([q+1,k,q-k+2]_q\) \textit{doubly extended narrow-sense RS code}.
   The two extra columns sometimes correspond to evaluating at zero and infinity if one switches to projective coordinates, in which case the doubly extended GRS code is an evaluation code.
+  There also exist \textit{triply extended RS codes} with parameters \([q+2,3,q-1]_q\) or \([q+2,q-1,4]_q\) \cite{doi:10.1201/9781315371993}.
 
   Their automorphism groups have been identified \cite{doi:10.1016/0097-3165(87)90060-4}.
 

codes/classical/q-ary_digits/ag/rs/generalized_reed_solomon.yml

@@ -34,7 +34,7 @@ features:
 realizations:
   - 'Commonly used in mass storage systems such as CDs, DVDs, QR codes etc.'
   - 'Various cloud storage systems \cite{arxiv:1612.01361}.'
-  - 'Public-key cryptosystems generalizing those that used Goppa codes \cite{manual:{R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Technical report, Jet Propulsion Lab. DSN Progress Report (1978).},doi:10.1007/BF00173300,manual:{H. Niederreiter (1986). \emph{Knapsack-type cryptosystems and algebraic coding theory}. Problems of Control and Information Theory. Problemy Upravlenija I Teorii Informacii. 15: 159–166.}}, some of which were proven to be insecure \cite{doi:10.1515/dma.1992.2.4.439}. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes \cite{doi:10.1007/s10623-003-6151-2}. For example, a key-recovery attack was developed in Ref. \cite{arxiv:1307.6458} for a variant of masking method proposed in Ref. \cite{arxiv:1108.2462}.'
+  - 'A variation of the McEliece public-key cryptosystem \cite{manual:{R. J. McEliece, A public-key cryptosystem based on algebraic coding theory, Technical report, Jet Propulsion Lab. DSN Progress Report (1978).},doi:10.1007/BF00173300} by Niederreiter \cite{manual:{H. Niederreiter (1986). \emph{Knapsack-type cryptosystems and algebraic coding theory}. Problems of Control and Information Theory. Problemy Upravlenija I Teorii Informacii. 15: 159–166.}} replaced the generator matrix by the parity check matrix of a GRS code. This was proven to be insecure since the public key exposes the algebraic structure of code \cite{doi:10.1515/dma.1992.2.4.439}. More recent works focus on methods to mask the algebraic structure using subcodes of GRS codes \cite{doi:10.1007/s10623-003-6151-2}. For example, a key-recovery attack was developed in Ref. \cite{arxiv:1307.6458} for a variant of masking method proposed in Ref. \cite{arxiv:1108.2462}.'
 
   #  - 'Generalized Reed Solomon codes contain the special case of conventiontal Reed Solomon codes. These are cyclic codes, and can also be defined by using the discrete Fourier transform.'
 

codes/classical/q-ary_digits/ag/varieties/deligne_lusztig.yml

@@ -11,12 +11,12 @@ name: 'Deligne-Lusztig code'
 introduced: '\cite{doi:10.1006/ffta.2001.0313,manual:{S.H. Hansen, The geometry of Deligne-Lusztig varieties: Higher dimensional AG codes, Ph.D. Thesis, University of Aarhus, 1999.},doi:10.1007/s002290050146,doi:10.1007/BFb0087993}'
 
 description: |
-  Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig variety.
+  Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig curve.
 
 relations:
   parents:
     - code_id: evaluation_polynomial
-      detail: 'Deligne-Lusztig codes are polynomial evaluation codes with \(\cal X\) a Deligne-Lusztig variety.'
+      detail: 'Deligne-Lusztig codes are evaluation AG codes with \(\cal X\) a Deligne-Lusztig curve.'
 
 
 # Begin Entry Meta Information

codes/classical/q-ary_digits/ag/varieties/grassmannian.yml

@@ -11,7 +11,7 @@ name: 'Grassmannian code'
 introduced: '\cite{manual:{C. T. Ryan, An application of Grassmannian varieties to coding theory. Congr. Numer. 57 (1987) 257–271.},manual:{C.T. Ryan, Projective codes based on Grassmann varieties, Congr. Numer. 57, 273–279 (1987).},doi:10.1016/0166-218X(90)90112-P}'
 
 description: |
-  Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\).
+  Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\) \cite{doi:10.1515/9783110811056.145}.
 
 relations:
   parents:

codes/classical/q-ary_digits/ag/varieties/serge.yml

@@ -7,16 +7,16 @@ code_id: serge
 physical: q-ary_digits
 logical: q-ary_digits
 
-name: 'Serge-variety RM-type code'
+name: 'Segre-variety RM-type code'
 introduced: '\cite{doi:10.1006/ffta.2002.0360}'
 
 description: |
-  Evaluation code of polynomials evaluated on points lying on a Serge variety.
+  Evaluation code of polynomials evaluated on points lying on a Segre variety.
 
 relations:
   parents:
     - code_id: evaluation_polynomial
-      detail: 'Serge-variety RM-type codes are polynomial evaluation codes with \(\cal X\) being a Serge variety.'
+      detail: 'Segre-variety RM-type codes are polynomial evaluation codes with \(\cal X\) being a Segre variety.'
 
 
 # Begin Entry Meta Information

codes/classical/q-ary_digits/easy/hexacode.yml

@@ -37,7 +37,7 @@ notes:
 relations:
   parents:
     - code_id: hyperoval
-      detail: 'Columns of hexacode''s generator matrix represent the six homogeneous coordinates of a hyperoval in the projective plane \(PG(2,4)\) \cite[pg. 289]{doi:10.1201/9781315371993}\cite[Exam. 19.2.1]{preset:HKStwoweight}.'
+      detail: 'Columns of hexacode''s generator matrix represent the six points of a hyperoval in the projective plane \(PG(2,4)\) \cite[pg. 289]{doi:10.1201/9781315371993}\cite[Exam. 19.2.1]{preset:HKStwoweight}.'
     - code_id: evaluation
       detail: 'The hexacode is an evaluation AG code over \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) \cite[Ex. 2.77]{preset:HPAlgCodes}.'
     - code_id: q-ary_quad_residue

codes/classical/q-ary_digits/group/cyclic/q-ary_bch.yml

@@ -12,7 +12,8 @@ short_name: 'BCH'
 introduced: '\cite{doi:10.1137/0109020}'
 
 description: |
-  Cyclic \(q\)-ary code, with \(n\) and \(q\) relatively coprime, whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\). More precisely, the generator polynomial of a BCH code of \textit{designed distance} \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called \textit{narrow-sense} when \(b=1\), and are called \textit{primitive} when \(n=q^r-1\) for some \(r\geq 2\).
+  Cyclic \(q\)-ary code, with \(n\) and \(q\) relatively prime, whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\). More precisely, the generator polynomial of a BCH code of \textit{designed distance} \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called \textit{narrow-sense} when \(b=1\), and are called \textit{primitive} when \(n=q^r-1\) for some \(r\geq 2\).
+  More general BCH codes can be defined for zeroes are powers of the form \(\{b,b+s,b+2s,\cdots,b+(\delta-2)s\}\) where gcd\((s,n)=1\).
 
   The code dimension is related to the \textit{multiplicative order} of \(q\) modulo \(n\), i.e., the smallest integer \(m\) such that \(n\) divides \(q^m-1\). The dimension of a BCH code is at least \(n-m(\delta-1)\). The field \(GF(q^m)\) is the smallest field containing the above root of unity \(\alpha\), and is the splitting field of the polynomial \(x^n-1\) (see \ref{topic:Cyclic-to-polynomial-correspondence}).
 
@@ -26,7 +27,6 @@ features:
     - 'Berlekamp-Massey decoder with runtime of order \(O(n^2)\) \cite{doi:10.1109/TIT.1968.1054109,doi:10.1109/TIT.1969.1054260,preset:Berlekamp} and modification by Burton \cite{doi:10.1109/TIT.1971.1054655}; see also \cite{preset:PetersonWeldon,doi:10.1007/978-3-7091-2945-6}.'
     - 'Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) \cite{doi:10.1109/TIT.1960.1057586,doi:10.1137/0109020} (see exposition in Ref. \cite{preset:Blahut}).'
     - 'Sugiyama et al. modification of the extended Euclidean algorithm \cite{doi:10.1016/S0019-9958(75)90090-X,doi:10.1017/CBO9780511606267}.'
-    - 'Guruswami-Sudan list decoder \cite{doi:10.1109/18.782097,doi:10.1109/SFCS.1998.743426} and modification by Koetter-Vardy for soft-decision decoding \cite{doi:10.1109/TIT.2003.819332}.'
 
 notes:
   - 'See books \cite{preset:MacSlo,preset:LinCostello,doi:10.1017/CBO9780511807077} for expositions on BCH codes and code tables.'