Best Known (93, 93+37, s)-Nets in Base 8
(93, 93+37, 1026)-Net over F8 — Constructive and digital
Digital (93, 130, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 65, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(93, 93+37, 4106)-Net over F8 — Digital
Digital (93, 130, 4106)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8130, 4106, F8, 37) (dual of [4106, 3976, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- linear OA(8129, 4097, F8, 37) (dual of [4097, 3968, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(8121, 4097, F8, 35) (dual of [4097, 3976, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
(93, 93+37, 3200187)-Net in Base 8 — Upper bound on s
There is no (93, 130, 3200188)-net in base 8, because
- 1 times m-reduction [i] would yield (93, 129, 3200188)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 315 216982 615821 096609 566263 395784 722493 657180 657867 309709 259733 601590 699706 651465 141527 077765 545456 374664 502828 725831 > 8129 [i]