MinT - Best Known (65−38, 65, s)-Nets in Base 256

Best Known (65−38, 65, s)-Nets in Base 256

(65−38, 65, 522)-Net over F₂₅₆ — Constructive and digital

Digital (27, 65, 522)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (4, 42, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆ (see above)

(65−38, 65, 1047)-Net over F₂₅₆ — Digital

Digital (27, 65, 1047)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(256⁶⁵, 1047, F₂₅₆, 38) (dual of [1047, 982, 39]-code), using
- 18 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 17 times 0) [i] based on linear OA(256⁶³, 1027, F₂₅₆, 38) (dual of [1027, 964, 39]-code), using
  - constructionÂ X applied to AG(F,985P) âŠ‚ AG(F,987P) [i] based on
    1. linear OA(256⁶², 1024, F₂₅₆, 38) (dual of [1024, 962, 39]-code), using algebraic-geometric code AG(F,985P) [i] based on function field F/F₂₅₆ with g(F) = 24 and N(F) ≥ 1025, using
      - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]
    2. linear OA(256⁶⁰, 1024, F₂₅₆, 36) (dual of [1024, 964, 37]-code), using algebraic-geometric code AG(F,987P) [i] based on function field F/F₂₅₆ with g(F) = 24 and N(F) ≥ 1025 (see above)
    3. linear OA(256¹, 3, F₂₅₆, 1) (dual of [3, 2, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(65−38, 65, 5387536)-Net in Base 256 — Upper bound on s

There is no (27, 65, 5387537)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 3 432398 910977 995301 586719 095926 877623 941355 809654 210158 185090 447063 399513 505156 677271 403592 509098 466828 293694 366853 852580 309328 145230 314712 350011 930592 297216 > 256⁶⁵ [i]