MinT - Best Known (37, 37+39, s)-Nets in Base 81

Best Known (37, 37+39, s)-Nets in Base 81

(37, 37+39, 730)-Net over F₈₁ — Constructive and digital

Digital (37, 76, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 76, 730)-net over F₈₁, using
- net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
    - the Hermitian function field over F₈₁ [i]

(37, 37+39, 1317)-Net over F₈₁ — Digital

Digital (37, 76, 1317)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁶, 1317, F₈₁, 39) (dual of [1317, 1241, 40]-code), using
- 4 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 0) [i] based on linear OA(81⁷⁵, 1312, F₈₁, 39) (dual of [1312, 1237, 40]-code), using
  - the BCH-code C(I) with length 1312Â | 81²âˆ’1, defining interval IÂ =Â {1,40,79,…,171}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]

(37, 37+39, 3385461)-Net in Base 81 — Upper bound on s

There is no (37, 76, 3385462)-net in base 81, because

1 times m-reduction [i] would yield (37, 75, 3385462)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 136892 158682 597651 128912 161549 419907 541737 443586 595469 276856 655090 184266 685572 194675 116630 399095 332301 088041 547409 773909 599901 942214 003237 057441 > 81⁷⁵ [i]