MinT - Best Known (12, 12+17, s)-Nets in Base 128

Best Known (12, 12+17, s)-Nets in Base 128

(12, 12+17, 342)-Net over F₁₂₈ — Constructive and digital

Digital (12, 29, 342)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (3, 20, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(12, 12+17, 392)-Net over F₁₂₈ — Digital

Digital (12, 29, 392)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128²⁹, 392, F₁₂₈, 17) (dual of [392, 363, 18]-code), using
- 7 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0) [i] based on linear OA(128²⁸, 384, F₁₂₈, 17) (dual of [384, 356, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
    1. linear OA(128²⁸, 382, F₁₂₈, 17) (dual of [382, 354, 18]-code), using an extension C_e(16) of the narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(128²⁶, 382, F₁₂₈, 16) (dual of [382, 356, 17]-code), using an extension C_e(15) of the narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(12, 12+17, 407)-Net in Base 128 — Constructive

(12, 29, 407)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. (3, 20, 257)-net in base 128, using
  - 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
    - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(12, 12+17, 513)-Net in Base 128

(12, 29, 513)-net in base 128, using

3 times m-reduction [i] based on (12, 32, 513)-net in base 128, using
- base change [i] based on digital (8, 28, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(12, 12+17, 703265)-Net in Base 128 — Upper bound on s

There is no (12, 29, 703266)-net in base 128, because

1 times m-reduction [i] would yield (12, 28, 703266)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 100434 707500 117272 012831 148179 459083 989168 837474 963143 368348 > 128²⁸ [i]