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Generic constructions of MDS Euclidean self-dual codes via GRS codes

  • * Corresponding author: Weijun Fang

    * Corresponding author: Weijun Fang 
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  • Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square $ q $, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of $ q $-ary MDS Euclidean self-dual codes of lengths in the form $ s\frac{q-1}{a}+t\frac{q-1}{b} $, where $ s $ and $ t $ range in some interval and $ a, b \,|\, (q -1) $. In particular, for large square $ q $, our constructions take up a proportion of generally more than 34% in all the possible lengths of $ q $-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.

    Mathematics Subject Classification: Primary: 94B05; Secondary: 94B60.

    Citation:

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  • Table 1.  Some known results about MDS self-dual codes with length n

    $ q $ $ n $ References
    $ q $ even $ n\leq q $ [12]
    $ q $ odd $ n=q+1 $ [12], [17]
    $ q=r^{2} $ $ n\leq r $ [17]
    $ q=r^{2} $, \; $ r\equiv3(mod\;4) $ $ n=2tr $, $ t\leq \frac{r-1}{2} $ [17]
    $ q\equiv1(mod\;4) $ $ 4^{n}n^{2}\leq q $ [17]
    $ q\equiv3(mod\;4) $ $ n\equiv 0 (mod\;4) $ and $ (n-1)\mid (q-1) $ [21]
    $ q\equiv1(mod\;4) $ $ (n-1)\mid (q-1) $ [21]
    $ q=p^{m}\equiv1(mod\;4) $ $ n=p^{l}+1 $, $ l\leq m $ [6]
    $ q=r^{s} $, $ r $ odd, $ s $ even $ n=2tr^{l} $, $ 0 \leq l\leq s $ and $ 1 \leq t \leq\frac{r-1}{2} $ [6]
    $ q=r^{s} $, $ r $ odd, $ s $ even $ n=(2t+1)r^{l}+1 $, $ 0 \leq l\leq s $ and $ 0 \leq t \leq\frac{r-1}{2} $ [6]
    $ q $ odd $ (n-2) \mid (q-1) $, $ \eta(2-n)=1 $ [22], [6]
    $ q $ odd $ (n-1) \mid (q-1) $, $ \eta(1-n)=1 $ [22]
    $ q\equiv1(mod\;4) $ $ n \mid (q-1) $ [22]
    $ q=p^{m} $, $ p $ odd $ n=p^{l}+1 $, $ l|m $ [22]
    $ q=p^{m} $, $ p $ odd $ n=2p^{l} $, $ l<m $, $ \eta(-1)=1 $ [22]
    $ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr $, $ t $ even and $ 2t \mid (r-1) $ [22]
    $ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr $, $ t $ even, $ (t-1) \mid (r-1) $ and $ \eta(1-t)=1 $ [22]
    $ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr+1 $, $ t $ odd, $ t \mid (r-1) $ and $ \eta(t)=1 $ [22]
    $ q=r^{s} $, $ r $ odd, $ s\geq2 $ $ n=tr+1 $, $ t $ odd, $ (t-1) \mid (r-1) $ and $ \eta(t-1)=\eta(-1)=1 $ [22]
    $ q=r^{2} $, $ r $ odd $ n=tr $, $ t $ even, $ 1\leq t\leq r $ [22]
    $ q=r^{2} $, $ r $ odd $ n=tr+1 $, $ t $ odd, $ 1\leq t\leq r $ [22]
    $ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $ [8]
    $ q=r^{2} $, $ r $ odd $ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $ and $ 2 \leq t \leq \frac{r+1}{2\gcd(r+1,m)} $ [8]
    $ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $ (except $ t $, $ m $ are even and $ r\equiv1(mod\;4) $), and $ 1 \leq t \leq \frac{r+1}{\gcd(r+1,m)} $ [8]
    $ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1),m)} $, $ s $ even, $ s \mid m $ and $ \frac{r+1}{s} $ even [8]
    $ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ \frac{q-1}{m} $ even, $ \frac{r+1}{s} $ even, $ 1 \leq t \leq \frac{s(r-1)}{\gcd(s(r-1), m)} $, $ s $ even and $ s \mid m $ [8]
    $ q=r^{2} $, $ r $ odd $ n=tm $, $ \frac{q-1}{m} $ even, $ 1 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
    $ q=r^{2} $, $ r $ odd $ n=tm+1 $, $ tm $ odd, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
    $ q=r^{2} $, $ r $ odd $ n=tm+2 $, $ tm $ even, $ m \mid (q-1) $, $ 2 \leq t \leq \frac{r-1}{\gcd(r-1,m)} $ [18]
    $ q=r^{2} $, $ r\equiv1(mod\;4) $ $ n=s(r-1)+t(r+1) $, $ s $ even, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $ [9]
    $ q=r^{2} $, $ r\equiv3(mod\;4) $ $ n=s(r-1)+t(r+1) $, $ s $ odd, $ 1 \leq s \leq \frac{r+1}{2} $ and $ 1 \leq t \leq \frac{r-1}{2} $ [9]
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    Table 2.  Proportion of number of possible lengths to $\frac{q}{2}$ ($N$ is the number of possible lengths)

    $r$ $q$ $N/(\frac{q}{2}$) of Table 1 (except [9]) $N/(\frac{q}{2})$ of [9] $N/\frac{q}{2}$ of us number of new lengths
    149 22201 $11.89\%$ $25\%$ $38.61\%$ 775
    151 22801 $13.16\%$ $25\%$ $34.95\%$ 676
    157 24649 $10.18\%$ $25\%$ $34.95\%$ 758
    163 26569 $10.67\%$ $25\%$ $34.28\%$ 828
    167 27889 $13.90\%$ $25\%$ $34.27\%$ 704
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