Best Known (48−17, 48, s)-Nets in Base 8
(48−17, 48, 354)-Net over F8 — Constructive and digital
Digital (31, 48, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 24, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
(48−17, 48, 516)-Net in Base 8 — Constructive
(31, 48, 516)-net in base 8, using
- base change [i] based on digital (19, 36, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 18, 258)-net over F256, using
(48−17, 48, 549)-Net over F8 — Digital
Digital (31, 48, 549)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(848, 549, F8, 17) (dual of [549, 501, 18]-code), using
- 29 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 18 times 0) [i] based on linear OA(844, 516, F8, 17) (dual of [516, 472, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(843, 512, F8, 17) (dual of [512, 469, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(840, 512, F8, 15) (dual of [512, 472, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(81, 4, F8, 1) (dual of [4, 3, 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 Ce(16) ⊂ Ce(14) [i] based on
- 29 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 18 times 0) [i] based on linear OA(844, 516, F8, 17) (dual of [516, 472, 18]-code), using
(48−17, 48, 578)-Net in Base 8
(31, 48, 578)-net in base 8, using
- base change [i] based on digital (19, 36, 578)-net over F16, using
- trace code for nets [i] based on digital (1, 18, 289)-net over F256, using
- net from sequence [i] based on digital (1, 288)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 1 and N(F) ≥ 289, using
- net from sequence [i] based on digital (1, 288)-sequence over F256, using
- trace code for nets [i] based on digital (1, 18, 289)-net over F256, using
(48−17, 48, 108699)-Net in Base 8 — Upper bound on s
There is no (31, 48, 108700)-net in base 8, because
- 1 times m-reduction [i] would yield (31, 47, 108700)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2 787716 096724 949287 315201 510073 002583 799131 > 847 [i]