simplex refs

codes/classical/q-ary_digits/projective/simplex.yml

@@ -8,17 +8,22 @@ physical: q-ary_digits
 logical: q-ary_digits
 
 name: 'Simplex code'
+introduced: '\cite{doi:10.1111/j.1469-1809.1941.tb02298.x,doi:10.1111/j.1469-1809.1943.tb02332.x}'
+#  from Simplex codes over the ring F_2+uF_2
 
 alternative_names:
   - 'Maximum-length feedback-shift-register code'
 
 description: |
-  An \([n,k,q^{k-1}]_q\) projective code with \(n=\frac{q^k-1}{q-1}\), denoted as \(S(q,k)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(k-1,q)\), with each column being a chosen representative of the corresponding element. Its dual code is the \([n,n-k,3]_q\) \(q\)-ary Hamming code. The name of the code comes from the property that, for \(q=2\), the codewords form a \((2^k-1)\)-simplex of constant edge length if the codewords are interpreted as points in \(\mathbb{R}^n\).
+  An \([n,k,q^{k-1}]_q\) projective code with \(n=\frac{q^k-1}{q-1}\), denoted as \(S(q,k)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(k-1,q)\), with each column being a chosen representative of the corresponding element. The name of the code comes from the property that, for \(q=2\), the codewords form a \((2^k-1)\)-simplex of constant edge length if the codewords are interpreted as points in \(\mathbb{R}^n\).
+
+  The dual of a simple code is the \([n,n-k,3]_q\) \(q\)-ary Hamming code.
+  A punctured simplex code is known as a \textit{MacDonald code} \cite{doi:10.1147/rd.41.0043}, with parameters \([[\frac{q^k-q^u}{q-1},k,q^{k-1}-q^{u-1}]]_q\) for \(u \leq k-1\) \cite{doi:10.1109/TIT.1975.1055315}.
 
 features:
   decoders:
-    - 'Due to the small size, it can be decoded according to maximum likelihood.'
-    - 'Some faster decoders for the \(q=2\) case: \cite{manual:{R. R. Green, "A serial orthogonal decoder," JPL Space Programs Summary, vol. 37–39-IV, pp. 247–253, 1966.},doi:10.1109/ITW.2003.1216684}'
+    - 'Permutation decoder for simplex \cite{doi:10.3934/amc.2012.6.505} and MacDonald \cite{doi:10.1007/s00200-016-0286-7} codes.'
+    - 'Decoders for the \(q=2\) case: \cite{manual:{R. R. Green, "A serial orthogonal decoder," JPL Space Programs Summary, vol. 37–39-IV, pp. 247–253, 1966.},doi:10.1109/ITW.2003.1216684}'
     - 'A quantum decoder for the \(q=2\) case: \cite{manual:{A. Barg and S. Zhou, “A quantum decoding algorithm for the simplex code”, in Proceedings of the 36th Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, 23–25 September 1998 (UIUC 1998) 359–365}}.'
 
 notes:
@@ -47,6 +52,8 @@ relations:
       detail: 'Binary simplex codes map to simplex spherical codes under the antipodal mapping \cite[Sec. 6.5.2]{manual:{Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.}}\cite[pg. 18]{preset:EricZin}. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.'
     - code_id: reed_muller
       detail: 'Binary simplex codes can be constructed from the generator matrix of RM\((1,k)\) by removing first the all-ones row, and then the all-zero column. Punctured RM codes and simplex codes are interconvertible via expurgation and augmentation (\cite{preset:MacSlo}, pg. 31).'
+    - code_id: two_weight
+      detail: 'MacDonald codes are the unique two-weight codes with weights \(q^{k-1}-q^{u-1}\) and \(q^{k-1}\) \cite{doi:10.1109/TIT.1975.1055315}.'
 
 
 # Begin Entry Meta Information