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Two classes of new optimal ternary cyclic codes

  • * Corresponding author: Xiwang Cao

    * Corresponding author: Xiwang Cao 

The first author is supported by the National Natural Science Foundation of China (No. 12001475) and the Natural Science Foundation of Jiangsu Province (No. BK20201059). The second author is supported by the National Natural Science Foundation of China (No. 12171241). The third author is supported by the National Natural Science Foundation of China (No. 11801070).

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  • Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes $ \mathcal{C}_{(2,v)} $ by using monomials $ x^2 $ and $ x^v $ for some suitable $ v $ and explain the novelty of the codes. Furthermore, the weight distribution of $ \mathcal{C}_{(2,v)}^{\perp} $ for $ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $ is determined.

    Mathematics Subject Classification: 94B15, 11T71.

    Citation:

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  • Table 1.  Value distribution of $ T(a,b) $

    Value Frequency
    0 $ (3^{m}-1)(3^{m}-2\cdot3^{m-1}+1) $
    $ 2\cdot3^{m} $ $ 1 $
    $ 3^{\frac{m+1}{2}} $ $ (3^{m}-1)(3^{m-1}+3^{\frac{m-1}{2}}) $
    $ - 3^{\frac{m+1}{2}} $ $ (3^{m}-1)(3^{m-1}-3^{\frac{m-1}{2}}) $
     | Show Table
    DownLoad: CSV

    Table 2.  Weight Distribution of $ \mathcal{C}_{(2,v)}^{\perp} $

    Weight Frequency
    $ 0 $ 1
    $ 3^{m}-3^{m-1}-3^{\frac{m-1}{2}} $ $ (3^{m}-1)(3^{m-1}+3^{\frac{m-1}{2}}) $
    $ 2\cdot 3^{m-1} $ $ (3^{m}-1)(3^{m}-2\cdot3^{m-1}+1) $
    $ 3^{m}-3^{m-1}+ 3^{\frac{m-1}{2}} $ $ (3^{m}-1)(3^{m-1}-3^{\frac{m-1}{2}}) $
     | Show Table
    DownLoad: CSV

    Table 3.  Known optimal ternary cyclic codes $ C_{1,e} $

    Conditions Case
    $ m $ is odd, $ e $ is even, $ e(3^{s}-1) \equiv 3^{t}-1 \pmod{ 3^{m}-1} $,
    $ 1\leq s,t\leq m-1 $, $ \gcd(m,s)=\gcd(m,t)=1 $, $ \gcd(3^{m}-1,e-1)=1 $.[22]
    $ 1) $
    $ e(3^{s}+1) \equiv 3^{t}+1 \pmod {3^{m}-1} $, $ 0\leq s,t\leq m-1 $, $ \gcd(3^{m}-1,e-1)= $
    $ \gcd(3^{m}-1,3^{t}-e)=1 $, $ m $ is either odd or even with $ \gcd(m,t)=1 $ and $ \frac{m}{\gcd(m,s)} $ is odd.[22]
    $ 2) $
    $ e \equiv \frac{3^{m}-1}{2}+3^{s}+1 \pmod {3^{m}-1} $, $ m $ is even, $ m/\gcd(m,s) $ is odd.[22] $ 3) $
    $ e \equiv \frac{3^{m}-1}{2}+3^{s}-1 \pmod {3^{m}-1} $, $ m $ is even, $ \gcd(m,s)= $
    $ \gcd(3^{s}-2,3^{m}-1)=1 $, $ s=1,3,5,7,9 $.[22]
    $ 4) $
    $ e = 3^{s}+5 $, $ m\equiv 0 \pmod 4 $ and $ s=\frac{m}{2} $ or $ m\equiv 2 \pmod 4 $ and $ s=\frac{m+2}{2} $.[10] $ 5) $
    $ m $ is odd and $ 1\leq s <m $, $ e = \frac{3^{s}+7}{2} $ if s is even, $ e =\frac{3^{m}-1}{2}+ \frac{3^{s}+7}{2} $
    if s is odd.[28]
    $ 6) $
    $ e = \frac{3^{\frac{m+1}{2}}+5}{2} $ if $ m \equiv 1 \pmod 4 $, $ e =\frac{3^{m}-1}{2}+ \frac{3^{\frac{m+1}{2}}+5}{2} $ if $ m \equiv 3 \pmod 4 $. [28] $ 7) $
    $ m $ is odd, $ e $ is even, $ e(3^{s}+1) \equiv \frac{3^{m}+1}{2} \pmod{ 3^{m}-1} $, $ 0\leq s \leq m-1 $.[28] $ 8) $
    $ 3 \nmid m $, $ 5e \equiv 2 \pmod{ 3^{m}-1} $.[28] $ 9) $
    $ 5 \nmid m $, $ 7e \equiv 2 \pmod{ 3^{m}-1} $, $ \gcd(m,6)=1 $ or $ m\equiv 3 \pmod{ 6} $.[28] $ 10) $
    $ m >2 $, $ 3 \nmid m $ and $ 5 \nmid m $, $ 5e \equiv 4 \pmod{ 3^{m}-1} $.[28] $ 11) $
     | Show Table
    DownLoad: CSV

    Table 4.  Known optimal ternary cyclic codes $ C_{1,e} $

    $ e $ Conditions Case
    2 $ m \geq 2 $[1] $ 12) $
    $ \frac{3^{s}+1}{2} $ $ s $ is odd, $ m \geq 2 $, $ \gcd(m,s)=1 $.[1] $ 13) $
    $ 3^{s}+1 $ $ m \geq 2 $, $ m/\gcd(m,s) $ is odd.[1] $ 14) $
    $ 3^{m-1}-1, \frac{3^{m}+1}{4}+\frac{3^{m}-1}{2} $ $ m \geq 3 $, $ m $ is odd.[4] $ 15) $
    $ 3^{\frac{m+1}{2}}-1 $ $ m $ is odd.[4] $ 16) $
    $ \frac{3^{m}-3}{2} $ $ m \geq 5 $, $ m $ is odd.[4] $ 17) $
    $ (3^{\frac{m+1}{4}}-1)(3^{\frac{m+1}{2}}+1) $ $ m \equiv 3 \pmod 4 $.[4] $ 18) $
    $ \frac{3^{(m+1)/2}-1}{2}+\frac{3^{m}-1}{2}, \frac{3^{m+1}-1}{8}+\frac{3^{m}-1}{2} $ $ m \equiv 1 \pmod 4 $.[4] $ 19) $
    $ \frac{3^{(m+1)/2}-1}{2}, \frac{3^{m+1}-1}{8} $ $ m \equiv 3 \pmod 4 $.[4] $ 20) $
    $ \frac{3^{s}-1}{2} $ $ m $ is odd, $ s $ is even,
    $ \gcd(m,s)=\gcd(m,s-1)=1 $.[4]
    $ 21) $
    $ 3^{s}-1 $ $ \gcd(m,s)=\gcd(3^{m}-1,3^{s}-2)=1 $.[4] $ 22) $
    $ \frac{3^{m-1}}{2}-2, \frac{3^{m-1}}{2}+10 $ $ m \equiv 2 \pmod{4} $.[13] $ 23) $
    $ \frac{3^{m-1}}{2}-5, \frac{3^{m-1}}{2}+7 $ $ m $ is odd.[13] $ 24) $
    $ 2(3^{m-1}-1), 5(3^{m-1}-1), 16 $ $ m $ is odd, $ 3\nmid m $.[13] $ 25) $
    $ 2(3^{s}+1) $ $ m $ is odd.[14] $ 26) $
     | Show Table
    DownLoad: CSV

    Table 5.  Known optimal ternary cyclic codes $ C_{u,v'} $

    $ u $ $ v' $ Conditions Case
    $ \frac{3^{m}+1}{2} $ $ \frac{3^{s}+1}{2} $ $ m $ is odd, $ s $ is even and $ \gcd(m,s)=1 $.[30] $ 27) $
    $ 2 \cdot 3^{\frac{m-1}{2}}+1 $ $ m $ is odd.[6] $ 28) $
    $ 3^{s}+2 $ $ m $ is odd, such that $ 9\nmid m $ and $ 4s \equiv1 \pmod m $.[25] $ 29) $
     | Show Table
    DownLoad: CSV
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