Best Known (147−42, 147, s)-Nets in Base 8
(147−42, 147, 1026)-Net over F8 — Constructive and digital
Digital (105, 147, 1026)-net over F8, using
- 7 times m-reduction [i] based on digital (105, 154, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 77, 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, 77, 513)-net over F64, using
(147−42, 147, 4134)-Net over F8 — Digital
Digital (105, 147, 4134)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8147, 4134, F8, 42) (dual of [4134, 3987, 43]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 27 times 0) [i] based on linear OA(8145, 4100, F8, 42) (dual of [4100, 3955, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(40) [i] based on
- linear OA(8145, 4096, F8, 42) (dual of [4096, 3951, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- 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]
- 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(41) ⊂ Ce(40) [i] based on
- 32 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 27 times 0) [i] based on linear OA(8145, 4100, F8, 42) (dual of [4100, 3955, 43]-code), using
(147−42, 147, 2600292)-Net in Base 8 — Upper bound on s
There is no (105, 147, 2600293)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 5 678433 555215 344635 808515 610645 315380 379233 149341 537474 499760 727546 029086 518437 734933 053317 312046 629562 724738 546931 801899 378382 744816 > 8147 [i]