nrt, poset + subspace as root code

codes/classical/properties/block/block.yml

@@ -35,6 +35,48 @@ description: |
 
   An alternative more stringent definition (not used here) is in terms of a map encoding logical information from \(\Sigma^k\) into \(\Sigma^n\), yielding an \((n,k,d)_{\Sigma}\) block code, where \(d\) is the code distance.
 
+  \subsection{Finite-field alphabet}
+  \begin{defterm}{Finite fields}
+  \label{topic:finite-fields}
+  The most common alphabets used in block codes Galois or finite fields \(GF(q)=\mathbb{F}_q\), which are sets of \(q\) elements closed under addition and multiplication.
+  They are finite analogues of the real or complex numbers, and a unique field exists for every power \(q=p^m\) of a prime \(p\).
+  The prime-field case reduces to \(\mathbb{Z}_p\), a group under addition that is promoted to a field by defining multiplication modulo \(p\); the case \(p=2\) yields the binary field \(\mathbb{Z}_2\).
+  Every finite field comes with a 0 element (additive identity), a 1 element (multiplicative identity), and additive (multiplicative) inverses for all (nonzero) elements.  
+  An element whose powers exhaust all nonzero field elements is called \textit{primitive}.
+  Fields come with a trace operation, the \textit{field trace}, which maps elements \(\gamma \in GF(q)\) to elements of \(GF(p)\) as
+  \begin{align}
+    \text{tr}(\gamma)=\sum_{k=0}^{m-1}\gamma^{p^{k}}~.
+  \end{align}
+  The field trace can be thought of as an averaging over the field's Galois group, which is the cyclic group generated by \(\gamma\to\gamma^p\) \cite[pg. 113]{preset:MacSlo}.
+  Fields also come with a \textit{field norm},
+  \begin{align}
+    N(\gamma)=\prod_{k=0}^{m-1}\gamma^{p^{k}}=\gamma^{(p^{m}-1)/(p-1)}~.
+  \end{align}
+  In the case of the complex numbers, analogues of the field trace and field norm are the real part and  norm squared of a complex number, respectively.
+
+  Any field \(GF(q)\) can be thought of as an \(m\)-dimensional vector space over \(GF(p)\) a.k.a. the \(m\)th \textit{extension} of \(GF(p)\) (similar to the complex numbers being an extension of the reals).
+  Conversely, \(GF(p)\) is an example of a \textit{subfield} of \(GF(q)\).
+  Certain field elements are chosen to be the \textit{basis} of \(GF(q)\) over \(GF(p)\), and all other elements are expressed as linear combinations of these basis elements.
+  More generally, elements of fields such as \(GF(p^{ml})\) can be written as \(m\)-dimensional vectors over \(GF(p^l)\) or \((m\times l)\)-dimensional matrices over \(GF(p)\). 
+  This idea is used to convert between ordinary block codes and matrix-based codes such as disk array codes and rank-metric codes.
+  The field norm and field trace can likewise be defined for fields \(GF(q^m)\) that are extensions of \(GF(q)\) for non-prime \(q\).
+  \end{defterm}
+
+  An example of a field is the quaternary Galois field \(GF(4) = \{0,1,\omega, \omega^2=\bar{\omega}\}\) with \(p=m=2\).
+  In this case, \(\omega\) can be interpreted as a third root of unity, but more formally it is defined as a solution to the polynomial equation \(1+x+x^2=0\).
+  Field elements can be represented as two-dimensional vectors with binary elements, \(GF(4)=GF(2^2)\), using the basis \(1\cong(1,0)\) and \(\omega\cong(0,1)\):
+  \begin{align}
+    0&\leftrightarrow(0,0)\cong0\cdot1+0\cdot\omega\\1&\leftrightarrow(0,1)\cong0\cdot1+1\cdot\omega\\\omega&\leftrightarrow(1,1)\cong1\cdot1+1\cdot\omega\\\bar{\omega}&\leftrightarrow(1,0)\cong1\cdot1+0\cdot\omega~.
+  \end{align}
+  In this way, the field elements form the Klein four group \(\mathbb{Z}_2\times\mathbb{Z}_2\) under addition.
+  One can check that the trace operation, \(\text{tr}(\gamma) = \gamma + \gamma^2\), outputs either 0 or 1 for any element \(\gamma\in GF(4)\).
+
+  Two \(q\)-ary codes are \textit{equivalent} if the codewords of one code can be mapped into those of the other under a combination of a coordinate permutation and a permutation of the elements of each coordinate. 
+  The full group of such composite permutations is \(S_q \wr S_n\) \cite[Def. 1.8.8]{preset:HKSbasics}\cite{preset:HKSclass}.
+
+# The field trace in this case is similar to taking the real part of a complex number, \(\text{tr}(\gamma) = \gamma + \gamma^2\) for any element \(\gamma\).
+
+# The group of isometries of \(q\)-ary Hamming space is a combination of the monomial group, the Galois group of \(GF(q)\), and the group formed by the action of the space on itself (under addition) \cite{preset:HKSclass}.
 
 # A common family of codes are the \textit{block codes}, intended to encode a piece, or block, of a data stream.
 # A block code encodes strings of length \(k\), where each character in the string an element of some fixed alphabet \(\Sigma\), into strings of length \(n\). In other words, a block code encoding is a map from \(\Sigma^k\) to \(\Sigma^n\), where \(N = |\Sigma|^n\), \(K=|\Sigma|^k\), and \(|\Sigma|\) is the number of elements in the alphabet.

codes/classical/q-ary_digits/insertion_deletion/insertion_deletion.yml → codes/classical/q-ary_digits/alternative_metrics/insertion_deletion/insertion_deletion.yml

@@ -31,6 +31,7 @@ notes:
   - 'See Refs. \cite{arxiv:1906.07887,doi:10.1109/TIT.2021.3056317}\cite[Sec. 22.7]{preset:HKSmetrics} for more details.'
 
 
+# Codewords are q-ary strings, so q-ary parent
 relations:
   parents:
     - code_id: q-ary_digits_into_q-ary_digits

codes/classical/q-ary_digits/alternative_metrics/poset/nrt.yml

@@ -0,0 +1,28 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: nrt
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: 'Niederreiter-Rosenbloom-Tsfasman (NRT) code'
+short_name: 'NRT code'
+introduced: '\cite{doi:10.1007/BF01294651,doi:10.1016/0012-365X(91)90315-S,doi:10.1016/0012-365X(92)90566-X,manual:{M. Yu. Rosenbloom, M. A. Tsfasman, “Codes for the m-Metric”, Probl. Peredachi Inf., 33:1 (1997), 55–63; Problems Inform. Transmission, 33:1 (1997), 45–52}}'
+
+description: |
+  A poset code based on the total ordering of \([n]\), i.e., \(1\leq 2\leq \cdots \leq n\).
+
+
+relations:
+  parents:
+    - code_id: poset
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-09-11'

codes/classical/q-ary_digits/alternative_metrics/poset/poset.yml

@@ -0,0 +1,43 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: poset
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: 'Poset code'
+introduced: '\cite{doi:10.1016/0012-365X(94)00228-B}'
+
+description: |
+  Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), with its distance evaluated in the poset metric.
+
+protection: |
+  Poset codes are quantified with respect to the poset metric \cite{preset:HKSmetrics}.
+  This metric is based on a partial ordering \(\leq\) on subsets of \([n]=\{1,2,\cdots,n\}\) and the notion of ideals generated by the support of an element of a \(q\)-ary string.
+  An ideal \(\langle \text{supp}(x) \rangle\) generated by \(x\in GF(q)^n\) contains all subsets of \([n]\) that are less than or equal to the subset in the support of \(x\) in the partial ordering.
+  The \textit{poset metric} between two strings \(x,y\) is then the cardinality of the ideal generated by the difference between the supports of \(x\) and \(y\), \(d_P(x,y) = |\langle \text{supp}(x-y) \rangle|\).
+
+  Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for poset codes, including generalizations of \hyperref[topic:weight-enumerator]{MacWilliams identities} \cite{arXiv:1205.1090}; see \cite{preset:HKSmetrics}.
+
+
+notes:
+    - 'See book \cite{doi:10.1007/978-3-319-93821-9} for more details.'
+
+
+# Codewords are q-ary strings, so q-ary parent
+relations:
+  parents:
+    - code_id: q-ary_digits_into_q-ary_digits
+  cousins:
+    - code_id: subspace
+      detail: 'Poset-code and subspace-code distance metric families intersect only at the Hamming metric \cite{preset:HKSmetrics}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-09-11'

codes/classical/q-ary_digits/alternative_metrics/subspace.yml

@@ -8,13 +8,13 @@ physical: q-ary_digits
 logical: q-ary_digits
 
 name: 'Subspace code'
-introduced: \cite{doi:10.1109/TIT.2008.926449}
+introduced: '\cite{doi:10.1109/TIT.2008.926449}'
 
 description: |
-  A code that is a set of subspaces of a projective space \(PG(n-1,q)\).
+  A code that is a set of subspaces of \(GF(q)^n\).
 
 protection: |
-  Subspace codes are quantified with respect to the subspace distance \cite{doi:10.1109/TIT.2008.926449} or injection distance \cite{arxiv:0805.3824}.'
+  Subspace codes are quantified with respect to the subspace distance \cite{doi:10.1109/TIT.2008.926449} or injection distance \cite{arxiv:0805.3824}.
   
   Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for subspace codes; see \cite{preset:HKSnetwork}.
 
@@ -25,9 +25,11 @@ features:
 realizations:
   - 'Packet-based transmission over networks \cite{preset:HKSnetwork}.'
 
+
+# 20240911: Codewords are subspaces, not elements of field, so q-ary is not parent
 relations:
   parents:
-    - code_id: q-ary_digits_into_q-ary_digits
+    - code_id: ecc_finite
   cousins:
     - code_id: projective
       detail: 'Subspace codes are sets of subspaces of a projective space \(PG(n-1,q)\).'

codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml

@@ -10,53 +10,12 @@ logical: q-ary_digits
 name: '\(q\)-ary code'
 
 description: |
-  Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\) and has distance \(d\). Usually denoted as \((n,K,d)_q\). The (Hamming) distance is the minimum number of coordinates where two strings in the code differ.
-
-  \subsection{Finite-field alphabet}
-  \begin{defterm}{Finite fields}
-  \label{topic:finite-fields}
-  Galois or finite fields \(GF(q)=\mathbb{F}_q\) are sets of \(q\) elements closed under addition and multiplication.
-  They are finite analogues of the real or complex numbers, and a unique field exists for every power \(q=p^m\) of a prime \(p\).
-  The prime-field case reduces to \(\mathbb{Z}_p\), a group under addition that is promoted to a field by defining multiplication modulo \(p\).
-  Every finite field comes with a 0 element (additive identity), a 1 element (multiplicative identity), and additive (multiplicative) inverses for all (nonzero) elements.  
-  An element whose powers exhaust all nonzero field elements is called \textit{primitive}.
-  Fields come with a trace operation, the \textit{field trace}, which maps elements \(\gamma \in GF(q)\) to elements of \(GF(p)\) as
-  \begin{align}
-    \text{tr}(\gamma)=\sum_{k=0}^{m-1}\gamma^{p^{k}}~.
-  \end{align}
-  The field trace can be thought of as an averaging over the field's Galois group, which is the cyclic group generated by \(\gamma\to\gamma^p\) \cite[pg. 113]{preset:MacSlo}.
-  Fields also come with a \textit{field norm},
-  \begin{align}
-    N(\gamma)=\prod_{k=0}^{m-1}\gamma^{p^{k}}=\gamma^{(p^{m}-1)/(p-1)}~.
-  \end{align}
-  In the case of the complex numbers, analogues of the field trace and field norm are the real part and  norm squared of a complex number, respectively.
-
-  Any field \(GF(q)\) can be thought of as an \(m\)-dimensional vector space over \(GF(p)\) a.k.a. the \(m\)th \textit{extension} of \(GF(p)\) (similar to the complex numbers being an extension of the reals).
-  Conversely, \(GF(p)\) is an example of a \textit{subfield} of \(GF(q)\).
-  Certain field elements are chosen to be the \textit{basis} of \(GF(q)\) over \(GF(p)\), and all other elements are expressed as linear combinations of these basis elements.
-  More generally, elements of fields such as \(GF(p^{ml})\) can be written as \(m\)-dimensional vectors over \(GF(p^l)\) or \((m\times l)\)-dimensional matrices over \(GF(p)\). 
-  This idea is used to convert between ordinary block codes and matrix-based codes such as disk array codes and rank-metric codes.
-  The field norm and field trace can likewise be defined for fields \(GF(q^m)\) that are extensions of \(GF(q)\) for non-prime \(q\).
-  \end{defterm}
-
-  An example of a field is the quaternary Galois field \(GF(4) = \{0,1,\omega, \omega^2=\bar{\omega}\}\) with \(p=m=2\).
-  In this case, \(\omega\) can be interpreted as a third root of unity, but more formally it is defined as a solution to the polynomial equation \(1+x+x^2=0\).
-  Field elements can be represented as two-dimensional vectors with binary elements, \(GF(4)=GF(2^2)\), using the basis \(1\cong(1,0)\) and \(\omega\cong(0,1)\):
-  \begin{align}
-    0&\leftrightarrow(0,0)\cong0\cdot1+0\cdot\omega\\1&\leftrightarrow(0,1)\cong0\cdot1+1\cdot\omega\\\omega&\leftrightarrow(1,1)\cong1\cdot1+1\cdot\omega\\\bar{\omega}&\leftrightarrow(1,0)\cong1\cdot1+0\cdot\omega~.
-  \end{align}
-  In this way, the field elements form the Klein four group \(\mathbb{Z}_2\times\mathbb{Z}_2\) under addition.
-  One can check that the trace operation, \(\text{tr}(\gamma) = \gamma + \gamma^2\), outputs either 0 or 1 for any element \(\gamma\in GF(4)\).
-
-  Two \(q\)-ary codes are \textit{equivalent} if the codewords of one code can be mapped into those of the other under a combination of a coordinate permutation and a permutation of the elements of each coordinate. 
-  The full group of such composite permutations is \(S_q \wr S_n\) \cite[Def. 1.8.8]{preset:HKSbasics}\cite{preset:HKSclass}.
-
-# The field trace in this case is similar to taking the real part of a complex number, \(\text{tr}(\gamma) = \gamma + \gamma^2\) for any element \(\gamma\).
-
-# The group of isometries of \(q\)-ary Hamming space is a combination of the monomial group, the Galois group of \(GF(q)\), and the group formed by the action of the space on itself (under addition) \cite{preset:HKSclass}.
+  Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), i.e., \(q\)-ary strings.
+  Their error-correcting performance is quantified by some distance \(d\), which in turn is defined using a metric.
+  The default distance is the Hamming distance \(d\), the weight (i.e., number of nonzero coordinates) of the lowest-weight nonzero codeword; such codes are usually denoted as \((n,K,d)_q\).
+  The corresponding Hamming metric between two \(q\)-ary strings is the number of coordinates in which they differ.
 
 protection: |
-  The standard metric for \(q\)-ary codes is the Hamming metric, but other metrics also exist \cite{arxiv:2212.00431,preset:HKSmetrics}.
   A code detects errors on up to \(d-1\) coordinates, corrects erasure errors on up to \(d-1\) coordinates, and corrects general errors on up to \(\left\lfloor (d-1)/2 \right\rfloor\) coordinates.
   Often, the relative distance \(\delta=d/n\) is used to compare codes of different lengths.
 

codetree/kingdoms.yml

@@ -12,6 +12,7 @@ kingdoms_by_domain_id:
       name: 'Galois-field Kingdom'
       root_codes:
         - code_id: q-ary_digits_into_q-ary_digits
+        - code_id: subspace
 
     - kingdom_id: matrices_into_matrices
       name: 'Matrix Kingdom'