Best Known (49−17, 49, s)-Nets in Base 8
(49−17, 49, 354)-Net over F8 — Constructive and digital
Digital (32, 49, 354)-net over F8, using
- 1 times m-reduction [i] based on digital (32, 50, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 25, 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
- trace code for nets [i] based on digital (7, 25, 177)-net over F64, using
(49−17, 49, 516)-Net in Base 8 — Constructive
(32, 49, 516)-net in base 8, using
- 81 times duplication [i] based on (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
- base change [i] based on digital (19, 36, 516)-net over F16, using
(49−17, 49, 591)-Net over F8 — Digital
Digital (32, 49, 591)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(849, 591, F8, 17) (dual of [591, 542, 18]-code), using
- 70 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 18 times 0, 1, 40 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
- 70 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 18 times 0, 1, 40 times 0) [i] based on linear OA(844, 516, F8, 17) (dual of [516, 472, 18]-code), using
(49−17, 49, 140967)-Net in Base 8 — Upper bound on s
There is no (32, 49, 140968)-net in base 8, because
- 1 times m-reduction [i] would yield (32, 48, 140968)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 22 301934 130549 421263 251113 288387 066009 173684 > 848 [i]