Best Known (129−36, 129, s)-Nets in Base 8
(129−36, 129, 1026)-Net over F8 — Constructive and digital
Digital (93, 129, 1026)-net over F8, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (28, 65, 513)-net over F64, using
(129−36, 129, 4294)-Net over F8 — Digital
Digital (93, 129, 4294)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 4294, F8, 36) (dual of [4294, 4165, 37]-code), using
- 190 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 43 times 0, 1, 138 times 0) [i] based on linear OA(8125, 4100, F8, 36) (dual of [4100, 3975, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- linear OA(8125, 4096, F8, 36) (dual of [4096, 3971, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8121, 4096, F8, 35) (dual of [4096, 3975, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- 190 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 43 times 0, 1, 138 times 0) [i] based on linear OA(8125, 4100, F8, 36) (dual of [4100, 3975, 37]-code), using
(129−36, 129, 3200187)-Net in Base 8 — Upper bound on s
There is no (93, 129, 3200188)-net in base 8, because
- 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]