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Character sums over a non-chain ring and their applications

  • * Corresponding author: Xiwang Cao

    * Corresponding author: Xiwang Cao

This work is supported by the National Natural Science Foundation of China under Grant 11771007

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  • Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.

    Mathematics Subject Classification: Primary: 11T24, 11L40, 11T71; Secondary: 11L05.

    Citation:

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  • Table 1.  Parameters of the $ (N_1, K_1) $ codebook of Theorem 5.1

    $ q $ $ (N_1, K_1) $ $ I_{\max}(C_1(R^*, \widehat{R}^*\times \widehat{R})) $ $ I_W $ $ \frac{I_{\max}(C_1(R^*, \widehat{R}^*\times \widehat{R}))}{I_W} $
    $ 13 $ $ (2028,144) $ 0.0902778 0.0803401 1.123695
    $ 23 $ $ (11638,484) $ 0.0475207 0.0445012 1.067852
    $ 5^2 $ $ (15000,576) $ 0.0434028 0.0408602 1.062227
    $ 7^2 $ $ (115248,2304) $ 0.0212674 0.0206241 1.031192
    $ 3^4 $ $ (524880,6400) $ 0.0126563 0.0124236 1.018730
    $ 103 $ $ (1082118,10404) $ 0.00990004 0.00975668 1.014694
    $ 151 $ $ (3420150,22500) $ 0.00666667 0.00664470 1.003311
    $ 5^4 $ $ (243750000,389376) $ 0.00160513 0.00160128 1.002404
    $ 7^5 $ $ (4.747279e+12,282441636) $ 0.0000595061 0.0000595008 1.000089
     | Show Table
    DownLoad: CSV

    Table 2.  Parameters of the $ (N_2, K_2) $ codebook of Theorem 5.2

    $ q $ $ (N_2, K_2) $ $ I_{\max}(C_2(R^*, \widehat{R}^*\times \widehat{R})) $ $ I_W $ $ \frac{I_{\max}(C_2(R^*, \widehat{R}^*\times \widehat{R}))}{I_W} $
    $ 17 $ $ (4352,256) $ 0.0664062 0.0606409 1.095074
    $ 37 $ $ (47952,1296) $ 0.0285494 0.0274001 1.041944
    $ 83 $ $ (558092,6724) $ 0.0123438 0.0121214 1.018347
    $ 11^2 $ $ (1742400,14400) $ 0.00840278 0.00829883 1.012526
    $ 3^5 $ $ (14231052,58564) $ 0.00414931 0.00412372 1.006205
    $ 293 $ $ (24982352,85264) $ 0.00343639 0.00341881 1.005141
    $ 7^3 $ $ (40118652,116964) $ 0.00293253 0.00291971 1.004389
    $ 13^3 $ $ (1.05948e+10,4822416) $ 0.000455581 0.00045527 1.000683
    $ 5^5 $ $ (3.0498e+10,9759376) $ 0.000320205 0.000320051 1.000480
     | Show Table
    DownLoad: CSV

    Table 3.  Parameters of the $ (N_3, K_3) $ codebook of Theorem 5.3

    $ q $ $ (N_3, K_3) $ $ I_{\max}({C}_3(D', \widehat{R}^*\times\widehat{R}^*)) $ $ I_W $ $ \frac{I_{\max}({C}_3(D', \widehat{R}^*\times\widehat{R}^*))}{I_W} $
    $ 19 $ $ (5832,324) $ 0.0657439 0.0573525 1.146314
    $ 59 $ $ (195112,3364) $ 0.0181594 0.0173972 1.043812
    $ 3^4 $ $ (512000,6400) $ 0.0129787 0.0125809 1.031622
    $ 113 $ $ (1404928,12544) $ 0.00917133 0.00896942 1.022511
    $ 211 $ $ (9261000,44100) $ 0.00483048 0.00477339 1.011959
    $ 281 $ $ (21952000,78400) $ 0.00360992 0.00357787 1.008959
    $ 5^4 $ $ (242970624,389376) $ 0.00161029 0.00160385 1.004013
    $ 11^3 $ $ (2.35264e+9,1768900) $ 0.000753578 0.000752163 1.001881
    $ 17^3 $ $ (1.18515e+11,24127744) $ 0.000203707 0.000203604 1.000509
     | Show Table
    DownLoad: CSV
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