MinT - Best Known (72, 72+53, s)-Nets in Base 16

Best Known (72, 72+53, s)-Nets in Base 16

(72, 72+53, 532)-Net over F₁₆ — Constructive and digital

Digital (72, 125, 532)-net over F₁₆, using

1 times m-reduction [i] based on digital (72, 126, 532)-net over F₁₆, using
- trace code for nets [i] based on digital (9, 63, 266)-net over F₂₅₆, using
  - net from sequence [i] based on digital (9, 265)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(72, 72+53, 1085)-Net over F₁₆ — Digital

Digital (72, 125, 1085)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁵, 1085, F₁₆, 53) (dual of [1085, 960, 54]-code), using
- 55 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 11 times 0, 1, 42 times 0) [i] based on linear OA(16¹²³, 1028, F₁₆, 53) (dual of [1028, 905, 54]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(16¹²², 1026, F₁₆, 53) (dual of [1026, 904, 54]-code), using
      - trace code [i] based on linear OA(256⁶¹, 513, F₂₅₆, 53) (dual of [513, 452, 54]-code), using
        extended algebraic-geometric code AG_e(F,459P) [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]
    2. linear OA(16¹²², 1027, F₁₆, 52) (dual of [1027, 905, 53]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16¹²²Â > V_b^sâˆ’1(kâˆ’1)Â = 647 541201 260657 039266 468286 788610 074332 352670 314324 999940 023904 646285 207970 657811 077217 014724 852149 279455 750443 066309 142089 083545 740416 644972 726016 [i]
    3. linear OA(16⁰, 1, F₁₆, 0) (dual of [1, 1, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(72, 72+53, 388966)-Net in Base 16 — Upper bound on s

There is no (72, 125, 388967)-net in base 16, because

1 times m-reduction [i] would yield (72, 124, 388967)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 204596 678393 374484 686459 420150 143551 802773 128303 114770 244290 925818 078744 852619 450797 585556 012828 157546 304123 059593 227098 173291 980652 179860 861086 672756 > 16¹²⁴ [i]