Best Known (31, 61, s)-Nets in Base 81
(31, 61, 452)-Net over F81 — Constructive and digital
Digital (31, 61, 452)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (16, 46, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- digital (0, 15, 82)-net over F81, using
(31, 61, 2189)-Net over F81 — Digital
Digital (31, 61, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8161, 2189, F81, 3, 30) (dual of [(2189, 3), 6506, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8161, 6567, F81, 30) (dual of [6567, 6506, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, 6569, F81, 30) (dual of [6569, 6508, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(8161, 6569, F81, 30) (dual of [6569, 6508, 31]-code), using
- OOA 3-folding [i] based on linear OA(8161, 6567, F81, 30) (dual of [6567, 6506, 31]-code), using
(31, 61, 4632851)-Net in Base 81 — Upper bound on s
There is no (31, 61, 4632852)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 261 569256 953909 843822 371968 125098 222151 681564 205951 594135 814158 758420 159582 952015 823758 487916 307487 367510 946079 678401 > 8161 [i]