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An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

  • *Corresponding author: René B. Christensen

    *Corresponding author: René B. Christensen 

This work was supported in part by Grant PGC2018-096446-B-C21 funded by MCIN/AEI/10.13039/501100011033 and by \ERDF A way of making Europe", by Grant RYC- 2016-20208 funded by MCIN/AEI/10.13039/501100011033 and by \ESF Investing in your future", and by the European Union's Horizon 2020 research and innovation programme, under grant agreement QUARTET No 862644

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  • We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $ c $, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $ c $ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.

    Mathematics Subject Classification: 94B65, 81P70, 94B05.

    Citation:

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  • Table 1.  Normalized reductions for $ q = 3 $ and $ m = 22 $

    $ i $ $ f_i $ $ \mathfrak{r}'(f_i^q) $ $ \mathfrak{r}(f_i^q)=\mathfrak{n}(\mathfrak{r}(f_i^q)) $
    $ 1 $ $ 1 $ $ 1 $ $ 1 $
    $ 2 $ $ x $ $ x^3 $ $ x^3 $
    $ 3 $ $ y $ $ x^4 $ $ + $ $ 2y $ $ x^4 $ $ + $ $ 2y $
    $ 4 $ $ x^2 $ $ x^6 $ $ x^6 $
    $ 5 $ $ xy $ $ x^7 $ $ + $ $ 2x^3y $ $ x^7 $ $ + $ $ 2x^3y $
    $ 6 $ $ y^2 $ $ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $ $ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $
    $ 7 $ $ x^3 $ $ x $ $ x $
    $ 8 $ $ x^2y $ $ 2x^6y $ $ + $ $ x^2 $ $ x^6y $ $ + $ $ 2x^2 $
    $ 9 $ $ xy^2 $ $ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $ $ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $
    $ 10 $ $ x^4 $ $ x^4 $ $ x^4 $
    $ 11 $ $ x^3y $ $ x^5 $ $ + $ $ 2xy $ $ x^5 $ $ + $ $ 2xy $
    $ 12 $ $ x^2y^2 $ $ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $ $ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $
    $ 13 $ $ x^5 $ $ x^7 $ $ x^7 $
    $ 14 $ $ x^4y $ $ x^8 $ $ + $ $ 2x^4y $ $ x^8 $ $ + $ $ 2x^4y $
    $ 15 $ $ x^3y^2 $ $ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $ $ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $
    $ 16 $ $ x^6 $ $ x^2 $ $ x^2 $
    $ 17 $ $ x^5y $ $ 2x^7y $ $ + $ $ x^3 $ $ x^7y $ $ + $ $ 2x^3 $
    $ 18 $ $ x^4y^2 $ $ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $ $ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $
    $ 19 $ $ x^7 $ $ x^5 $ $ x^5 $
    $ 20 $ $ x^6y $ $ x^6 $ $ + $ $ 2x^2y $ $ x^6 $ $ + $ $ 2x^2y $
     | Show Table
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    Table 2.  Algorithm 1 for $ q = 3 $

    $ f_i $ $ \nu(f_i) $ $ \mathfrak{r}(f_i^q) $ $ \nu(\mathfrak{r}(f_i^q)) $ $ \phi_i $ $ \nu(\phi_i) $
    $ 1 $ 0 $ 1 $ 0 $ 1 $ 0
    $ x $ 3 $ x^3 $ 9 $ x^3 $ 9
    $ y $ 4 $ x^4+2y $ 12 $ x^4+2y $ 12
    $ x^2 $ 6 $ x^6 $ 18 $ x^6 $ 18
    $ xy $ 7 $ x^7+2x^3y $ 21 $ x^7+2x^3y $ 21
    $ y^2 $ 8 $ x^8+x^4y+y^2 $ 24 $ x^8+x^4y+y^2 $ 24
    $ x^3 $ 9 $ x $ 3 $ x $ 3
    $ x^2y $ 10 $ x^6y+2x^2 $ 22 $ x^6y+2x^2 $ 22
    $ xy^2 $ 11 $ x^7y+x^3y^2+x^3 $ 25 $ x^7y+x^3y^2+x^3 $ 25
    $ x^4 $ 12 $ x^4 $ 12 $ y $ 4
    $ x^3y $ 13 $ x^5+2xy $ 15 $ x^5+2xy $ 15
    $ x^2y^2 $ 14 $ x^6y^2+x^6+x^2y $ 26 $ x^6y^2+x^6+x^2y $ 26
    $ x^5 $ 15 $ x^7 $ 21 $ x^3y $ 13
     | Show Table
    DownLoad: CSV

    Table 3.  Examples of code's parameters and comparative analysis by means of coding bounds

    Parameters Singleton defect Exceeding GV
    $ [[27, 1, 19; 16]]_3 $ 6
    $ [[27, 4, 16; 13]]_3 $ 6
    $ [[27, 13, 7; 4]]_3 $ 6
    $ [[27, 16, 4; 1]]_3 $ 6
    $ [[64, 5, 42; 35]]_4 $ 12
    $ [[64, 16, 30; 22]]_4 $ 12
    $ [[64, 35, 12; 3]]_4 $ 10
    $ [[64, 39, 8; 1]]_4 $ 12
    $ [[125, 1,101; 96]]_5 $ 20
    $ [[125, 9, 91; 84]]_5 $ 20
    $ [[125, 36, 56; 41]]_5 $ 20
    $ [[125, 70, 26; 15]]_5 $ 20
    $ [[125, 90, 10; 1]]_5 $ 18
     | Show Table
    DownLoad: CSV

    Table 4.  Monomials in the support of $ \mathfrak{r}(f_{24}^q) $ for $ q = 5 $

    $ j $ $ f_j $ $ \nu(f_j) $ $ \nu(f_j)\bmod{(q^2-1)} $
    $ 108 $ $ x^{21}y^2 $ $ 117 $ $ 21 $
    $ 84 $ $ x^{15}y^3 $ $ 93 $ $ 21 $
    $ 36 $ $ x^9 $ $ 45 $ $ 21 $
    $ 12 $ $ x^3y $ $ 21 $ $ 21 $
     | Show Table
    DownLoad: CSV

    Table 5.  Results of the modified algorithm for $ q = 3 $

    $ f_i $ $ \nu(f_i) $ $ \mathfrak{r}(f_i^q) $ $ \nu(\mathfrak{r}(f_i^q)) $ $ \phi_i $ $ \nu(\phi_i) $
    $ 1 $ 0 0 0
    $ x $ 3 9 9
    $ y $ 4 $ x^4+2y $ 12 $ x^4+2y $ 12
    $ x^2 $ 6 18 18
    $ xy $ 7 $ x^7+2x^3y $ 21 $ x^7+2x^3y $ 21
    $ y^2 $ 8 24 24
    $ x^3 $ 9 3 3
    $ x^2y $ 10 22 22
    $ xy^2 $ 11 25 25
    $ x^4 $ 12 $ x^4 $ 12 $ y $ 4
    $ x^3y $ 13 15 15
    $ x^2y^2 $ 14 26 26
    $ x^5 $ 15 $ x^7 $ 21 $ x^3y $ 13
     | Show Table
    DownLoad: CSV

    Table 6.  Hermitian EAQECCs exceding the GV bound

    $ q $ $ n $ $ c $
    2 8 0–3
    3 27 1–16
    4 64 3–45
    5 125 4–96
    7 343 10–288
    8 512 9–441
    9 729 14–640
    11 1331 38–1200
    13 2197 51–2016
    16 4096 45–3825
     | Show Table
    DownLoad: CSV
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