MinT - Best Known (258−127, 258, s)-Nets in Base 2

Best Known (258−127, 258, s)-Nets in Base 2

(258−127, 258, 57)-Net over F₂ — Constructive and digital

Digital (131, 258, 57)-net over F₂, using

t-expansion [i] based on digital (110, 258, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(258−127, 258, 81)-Net over F₂ — Digital

Digital (131, 258, 81)-net over F₂, using

t-expansion [i] based on digital (126, 258, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(258−127, 258, 280)-Net over F₂ — Upper bound on s (digital)

There is no digital (131, 258, 281)-net over F₂, because

1 times m-reduction [i] would yield digital (131, 257, 281)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁷, 281, F₂, 126) (dual of [281, 24, 127]-code), but
  - residual code [i] would yield linear OA(2¹³¹, 154, F₂, 63) (dual of [154, 23, 64]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(258−127, 258, 291)-Net in Base 2 — Upper bound on s

There is no (131, 258, 292)-net in base 2, because

19 times m-reduction [i] would yield (131, 239, 292)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁹, 292, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 129 753212 264703 737764 759763 826530 014325 194921 459040 258227 297024 189374 782768 881401 211050 262528 / 138 951361 618522 578125 > 2²³⁹ [i]