q-ary lcc

codes/classical/bits/easy/hadamard.yml

@@ -19,9 +19,7 @@ relations:
       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: balanced
       detail: 'Each Hadamard codeword has length \(2^m\) and Hamming weight of \(2^{m-1}\), making this code balanced.'
-    - code_id: ldc
-      detail: 'Hadamard codes are two-query LDCs and LCCs \cite{doi:10.1561/0400000030}; see Ref. \cite[Sec. 2.4.1]{manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.'
-    - code_id: lcc
+    - code_id: q-ary_lcc
       detail: 'Hadamard codes are two-query LDCs and LCCs \cite{doi:10.1561/0400000030,manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.'
     - code_id: binary_ltc
       detail: 'The Hadamard code is the first code to be identified as a (three-query) LDC \cite{doi:10.1145/100216.100225,doi:10.1016/0022-0000(93)90044-W}.'

codes/classical/properties/block/distributed_storage/lrc/lcc.yml

@@ -20,13 +20,7 @@ description: |
   Recall that a block code encodes a length-\(k\) message into a length-\(n\) codeword, which is then sent through a noise channel to yield an error word.
   Informally, an LCC is a block code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of the error word (assuming the error word is within the decoding radius).
 
-  LDCs have applications in computational complexity theory and cryptography \cite[Sec. 17.4]{doi:10.1017/CBO9780511804090}\cite{doi:10.1561/0400000030,doi:10.1145/301250.301397,doi:10.1145/2993749.2993761}.
-
-  Modified versions of local decodability include \textit{relaxed local decodability} \cite{doi:10.1145/1007352.1007361}.
-
-features:
-  decoders:
-    - 'LDCs admit \textit{local decoders}, i.e., decoders whose runtime scales polylogarithmically with \(n\).'
+  Modified versions of local correctability include \textit{relaxed local correctability} \cite{doi:10.4086/toc.2020.v016a018}.
 
 
 relations:
@@ -36,8 +30,7 @@ relations:
       Since a codeword is a trivial error word, any LCC is also an LRC.'
   cousins:
     - code_id: ldc
-      detail: 'Linear LCCs are automatically LDCs \cite[Sec. 2.4.1]{manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.
-      More generally, any family of LCCs can be converted to a family of LDCs whose rate differs by a constant \cite{arxiv:1611.06980}.'
+      detail: 'Any family of LCCs can be converted to a family of LDCs whose rate differs by a constant \cite{arxiv:1611.06980}; see \cite[Sec. 2.4.1]{manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.'
     - code_id: ltc
       detail: 'There are relations between LDCs and LTCs \cite{doi:10.1007/978-3-642-15369-3_50}.'
     - code_id: quantum_locally_recoverable
@@ -49,4 +42,4 @@ _meta:
   # Change log - most recent first
   changelog:
     - user_id: VictorVAlbert
-      date: '2023-05-05'
+      date: '2023-03-27'

codes/classical/q-ary_digits/ag/generalized_reed_muller.yml

@@ -33,15 +33,14 @@ relations:
       detail: 'Multivariate multiplicity codes of degree \(s=1\) reduce to GRM codes.'
     - code_id: matrix_product
       detail: 'Applying a special case of the matrix-product procedure yields GRM codes \cite{doi:10.1007/PL00004226}.'
-    - code_id: ldc
-      detail: 'GRM codes are LDCs and LCCs \cite{doi:10.1561/0400000030,manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.'
-    - code_id: lcc
+    - 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: q-ary_cyclic
       detail: 'GRM codes with nonzero evaluation points are cyclic (\cite{doi:10.1007/978-94-011-3810-9}, pg. 52).'
     - code_id: q-ary_ltc
-      detail: 'GRM codes for \(r<q\) can be LTCs in the low- \cite{doi:10.1145/103418.103428,doi:10.1145/278298.278306} and high-error \cite{doi:10.1145/258533.258641,doi:10.1145/258533.258642} regimes. They admit weakly stable presentations of their correspoding groups \cite{arxiv:2311.04681}.'
+      detail: 'GRM codes for \(r<q\) can be LTCs in the low- \cite{doi:10.1145/103418.103428,doi:10.1145/278298.278306} and high-error \cite{doi:10.1145/258533.258641,doi:10.1145/258533.258642} regimes.
+      They admit weakly stable presentations of their correspoding groups \cite{arxiv:2311.04681}.'
     - 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}.'
 

codes/classical/q-ary_digits/ag/multiplicity.yml

@@ -35,9 +35,7 @@ relations:
     - code_id: group
       detail: 'Multiplicity codes of order \(s\) are Abelian group-algebra codes whose corresponding polynomial that is modded out is \((x-\alpha_j)^s\) for each evaluation point \(\alpha_j\) \cite{arxiv:2103.07930}.'
   cousins:
-    - code_id: ldc
-      detail: 'There exist multiplicity codes with rate arbitrarily close to one that are locally decodable and locally correctable from a constant error fraction \cite{doi:10.1145/2629416}.'
-    - code_id: lcc
+    - code_id: q-ary_lcc
       detail: 'There exist multiplicity codes with rate arbitrarily close to one that are locally decodable and locally correctable from a constant error fraction \cite{doi:10.1145/2629416}.'
     - code_id: ltc
       detail: 'Some multiplicity codes are locally testable by an appropriate test \cite{doi:10.4230/LIPIcs.CCC.2022.14,doi:10.4230/LIPIcs.APPROX/RANDOM.2022.11}.'

codes/classical/q-ary_digits/distributed_storage/q-ary_lcc.yml

@@ -0,0 +1,29 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: q-ary_lcc
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: '\(q\)-ary linear LCC'
+
+
+description: |
+  A linear code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of the error word (assuming the error word is within the decoding radius).
+
+relations:
+  parents:
+    - code_id: q-ary_linear
+    - code_id: lcc
+    - code_id: ldc
+      detail: 'Linear LCCs can be converted into LDCs with the same locality \(r\) \cite[Sec. 2.4.1]{manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2023-03-27'