MinT - Best Known (26−15, 26, s)-Nets in Base 32

Best Known (26−15, 26, s)-Nets in Base 32

(26−15, 26, 120)-Net over F₃₂ — Constructive and digital

Digital (11, 26, 120)-net over F₃₂, using

net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
  - a function field by SÃ©mirat [i]

(26−15, 26, 130)-Net over F₃₂ — Digital

Digital (11, 26, 130)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²⁶, 130, F₃₂, 15) (dual of [130, 104, 16]-code), using
- 28 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 6 times 0, 1, 19 times 0) [i] based on linear OA(32²², 98, F₃₂, 15) (dual of [98, 76, 16]-code), using
  - extended algebraic-geometric code AG_e(F,82P) [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(26−15, 26, 257)-Net in Base 32 — Constructive

(11, 26, 257)-net in base 32, using

2 times m-reduction [i] based on (11, 28, 257)-net in base 32, using
- base change [i] based on (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]

(26−15, 26, 25884)-Net in Base 32 — Upper bound on s

There is no (11, 26, 25885)-net in base 32, because

1 times m-reduction [i] would yield (11, 25, 25885)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 42 541943 287809 145623 942988 331964 590168 > 32²⁵ [i]