Best Known (37, 75, s)-Nets in Base 81
(37, 75, 730)-Net over F81 — Constructive and digital
Digital (37, 75, 730)-net over F81, using
- t-expansion [i] based on digital (36, 75, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(37, 75, 1845)-Net over F81 — Digital
Digital (37, 75, 1845)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8175, 1845, F81, 3, 38) (dual of [(1845, 3), 5460, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8175, 2187, F81, 3, 38) (dual of [(2187, 3), 6486, 39]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8175, 6561, F81, 38) (dual of [6561, 6486, 39]-code), using
- an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- OOA 3-folding [i] based on linear OA(8175, 6561, F81, 38) (dual of [6561, 6486, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(8175, 2187, F81, 3, 38) (dual of [(2187, 3), 6486, 39]-NRT-code), using
(37, 75, 3385461)-Net in Base 81 — Upper bound on s
There is no (37, 75, 3385462)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 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]