Best Known (147−41, 147, s)-Nets in Base 8
(147−41, 147, 1026)-Net over F8 — Constructive and digital
Digital (106, 147, 1026)-net over F8, using
- 9 times m-reduction [i] based on digital (106, 156, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 78, 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, 78, 513)-net over F64, using
(147−41, 147, 4715)-Net over F8 — Digital
Digital (106, 147, 4715)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8147, 4715, F8, 41) (dual of [4715, 4568, 42]-code), using
- 613 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 43 times 0, 1, 129 times 0, 1, 199 times 0, 1, 231 times 0) [i] based on linear OA(8141, 4096, F8, 41) (dual of [4096, 3955, 42]-code), using
- an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- 613 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 43 times 0, 1, 129 times 0, 1, 199 times 0, 1, 231 times 0) [i] based on linear OA(8141, 4096, F8, 41) (dual of [4096, 3955, 42]-code), using
(147−41, 147, 4642628)-Net in Base 8 — Upper bound on s
There is no (106, 147, 4642629)-net in base 8, because
- 1 times m-reduction [i] would yield (106, 146, 4642629)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 709804 747650 937316 176237 879013 478345 424516 557535 608448 272688 056753 602950 499535 689561 079857 386881 105102 446404 224877 702805 913074 967196 > 8146 [i]