MinT - Best Known (122, 122+129, s)-Nets in Base 2

Best Known (122, 122+129, s)-Nets in Base 2

(122, 122+129, 57)-Net over F₂ — Constructive and digital

Digital (122, 251, 57)-net over F₂, using

t-expansion [i] based on digital (110, 251, 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]

(122, 122+129, 80)-Net over F₂ — Digital

Digital (122, 251, 80)-net over F₂, using

t-expansion [i] based on digital (121, 251, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(122, 122+129, 255)-Net over F₂ — Upper bound on s (digital)

There is no digital (122, 251, 256)-net over F₂, because

1 times m-reduction [i] would yield digital (122, 250, 256)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁰, 256, F₂, 128) (dual of [256, 6, 129]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2²⁵¹, 257, F₂, 128) (dual of [257, 6, 129]-code), but
    - Griesmer bound [i]

(122, 122+129, 260)-Net in Base 2 — Upper bound on s

There is no (122, 251, 261)-net in base 2, because

21 times m-reduction [i] would yield (122, 230, 261)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁰, 261, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 2423 906907 580468 277479 349413 556067 053430 053216 032374 525434 104378 793213 098031 316992 / 1 326150 802185 > 2²³⁰ [i]