Geodesic
Ternary linear codes and quadrics
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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For an $[n,k,d]_3$ code ${\cal C}$ with $gcd(d,3)=1$, we define a map $w_G$ from $\Sigma={\rm PG}(k-1,3)$ to the set of weights of codewords of ${\cal C}$ through a generator matrix $G$. A $t$-flat $\Pi$ in $\Sigma$ is called an $(i,j)_t$ flat if $(i,j)=(|\Pi \cap F_0|,|\Pi \cap F_1|)$, where $F_0 = \{P \in \Sigma | w_G(P) \equiv 0 \pmod{3}\}$, $F_1 = \{P \in \Sigma | w_G(P) \not\equiv 0,d \pmod{3}\}$. We give geometric characterizations of $(i,j)_t$ flats, which involve quadrics. As an application to the optimal linear codes problem, we prove the non-existence of a $[305,6,202]_3$ code, which is a new result.
DOI : 10.37236/98
Classification : 94B27, 94B05, 51E20, 05B25 
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     author = {Yuri Yoshida and Tatsuya Maruta},
     title = {Ternary linear codes and quadrics},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {1},
     doi = {10.37236/98},
     zbl = {1160.94016},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/98/}
}
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Yuri Yoshida; Tatsuya Maruta. Ternary linear codes and quadrics. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/98

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