~

codes/classical/bits/covering/nearly_perfect.yml

@@ -16,7 +16,7 @@ description: |
     \frac{{n \choose t}\left(\frac{n-t}{t+1}-\left\lfloor \frac{n-t}{t+1}\right\rfloor \right)}{\left\lfloor \frac{n}{t+1}\right\rfloor }+\sum_{j=0}^{t}{n \choose j}\leq2^{n}/K
   \end{align}
   becomes an equality (\cite{doi:10.1017/CBO9780511807077}, Sec. 2.3.5; see also Ref. \cite{preset:MacSlo}, Ch. 17).
-  All nearly perfect binary codes are either perfect, or correspond to either punctured Preparata codes or one of the \(2^{2^r-2-r},2^r-2,3)\) codes for \(r\geq 3\) \cite{manual:{Kauko Lindström. "The nonexistence of unknown nearly perfect binary codes." PhD diss., Turun yliopisto, 1975.}}.
+  All nearly perfect binary codes are either perfect, or correspond to either Preparata codes or one of the \((2^{2^r-2-r},2^r-2,3)\) codes for \(r\geq 3\) \cite{manual:{Kauko Lindström. "The nonexistence of unknown nearly perfect binary codes." PhD diss., Turun yliopisto, 1975.}}.
 
   Similar definitions can be made for \(q\)-ary codes, but all nearly perfect \(q\)-ary codes must be perfect \cite{manual:{K. Lindstrom and M. J. Aaltonen, "The nonexistence of nearly perfect nonbinary codes for 1 =< e =< 10", Ann. Univ. Turku, Ser. A I, No. 172, 1976.},doi:10.1016/S0019-9958(77)90519-8}.
 

codes/classical/bits/nonlinear/preparata.yml

@@ -35,11 +35,12 @@ notes:
 relations:
   parents:
     - code_id: nearly_perfect
-      detail: 'Preparata codes are uniformly packed, quasi-perfect \cite[pg. 475]{preset:MacSlo}, and nearly perfect.
-      For any word \(u\) and Preparata codebook \(C\) with \(d(u, C) > 2\), we have that \(u\) has a distance 2 or 3 to exactly floor((2^{m+1}-1)/3) codewords.'
+      detail: 'Preparata codes are uniformly packed and nearly perfect \cite{doi:10.1016/0012-365X(72)90025-8}. For any word \(u\) and Preparata codebook \(C\) with \(d(u, C) > 2\), we have that \(u\) has a distance 2 or 3 to exactly floor((2^{m+1}-1)/3) codewords.'
   cousins:
     - code_id: quaternary_over_z4
       detail: 'Preparata codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
+    - code_id: quasi_perfect
+      detail: 'Punctured Preparata codes are quasi-perfect \cite[pg. 475]{preset:MacSlo}.'
     - code_id: reed_muller
       detail: 'A Preparata code can be written as a union of a linear subcode \(\mathcal{C}\) of RM\((m-2,m)\) and the \(2^{m-1}-1\) representatives of coset formed by \(\mathcal{C}\) in RM\((m-2,m)\). The coset representatives are given by \(|1|x^j|0|x^{j}\theta_{1}|\), where \(1\leq j \leq 2^{m-1}-1\). \(\mathcal{C}\) comprises of codewords of the form \(|g(1)|g(x)(1+\theta_{1})|f(1)+g(1)|g(x)(1+\theta_{1})+f(x)(1+\theta_{1}+\theta_{3})|\), where \(f(x)\) and \(g(x)\) are arbitrary, and where \(\theta_{1}\) and \(\theta_{3}\) denote the primitive idempotents corresponding to cyclotomic cosets \(C_1\) and \(C_3\) respectively.'
     - code_id: bch