Best Known (76−38, 76, s)-Nets in Base 81
(76−38, 76, 730)-Net over F81 — Constructive and digital
Digital (38, 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
- 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
(76−38, 76, 2102)-Net over F81 — Digital
Digital (38, 76, 2102)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8176, 2102, F81, 3, 38) (dual of [(2102, 3), 6230, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8176, 2188, F81, 3, 38) (dual of [(2188, 3), 6488, 39]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8176, 6564, F81, 38) (dual of [6564, 6488, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(8176, 6566, F81, 38) (dual of [6566, 6490, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(35) [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]
- linear OA(8171, 6561, F81, 36) (dual of [6561, 6490, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(37) ⊂ Ce(35) [i] based on
- discarding factors / shortening the dual code based on linear OA(8176, 6566, F81, 38) (dual of [6566, 6490, 39]-code), using
- OOA 3-folding [i] based on linear OA(8176, 6564, F81, 38) (dual of [6564, 6488, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(8176, 2188, F81, 3, 38) (dual of [(2188, 3), 6488, 39]-NRT-code), using
(76−38, 76, 4266430)-Net in Base 81 — Upper bound on s
There is no (38, 76, 4266431)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 11 088244 582578 440216 118969 874628 326810 474918 268083 070098 862766 514931 622250 602145 451641 535026 877542 501901 759268 196042 888990 824580 853280 329070 165521 > 8176 [i]