Best Known (79−36, 79, s)-Nets in Base 81
(79−36, 79, 730)-Net over F81 — Constructive and digital
Digital (43, 79, 730)-net over F81, using
- t-expansion [i] based on digital (36, 79, 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
(79−36, 79, 4027)-Net over F81 — Digital
Digital (43, 79, 4027)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8179, 4027, F81, 36) (dual of [4027, 3948, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8179, 6587, F81, 36) (dual of [6587, 6508, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(26) [i] based on
- 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(8153, 6561, F81, 27) (dual of [6561, 6508, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(818, 26, F81, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,81)), using
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- Reed–Solomon code RS(73,81) [i]
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- construction X applied to Ce(35) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(8179, 6587, F81, 36) (dual of [6587, 6508, 37]-code), using
(79−36, 79, large)-Net in Base 81 — Upper bound on s
There is no (43, 79, large)-net in base 81, because
- 34 times m-reduction [i] would yield (43, 45, large)-net in base 81, but