Best Known (159, 159+36, s)-Nets in Base 4
(159, 159+36, 1539)-Net over F4 — Constructive and digital
Digital (159, 195, 1539)-net over F4, using
- t-expansion [i] based on digital (158, 195, 1539)-net over F4, 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
(159, 159+36, 12265)-Net over F4 — Digital
Digital (159, 195, 12265)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4195, 12265, F4, 36) (dual of [12265, 12070, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4195, 16410, F4, 36) (dual of [16410, 16215, 37]-code), using
- strength reduction [i] based on linear OA(4195, 16410, F4, 37) (dual of [16410, 16215, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(32) [i] based on
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(36) ⊂ Ce(32) [i] based on
- strength reduction [i] based on linear OA(4195, 16410, F4, 37) (dual of [16410, 16215, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4195, 16410, F4, 36) (dual of [16410, 16215, 37]-code), using
(159, 159+36, 8381555)-Net in Base 4 — Upper bound on s
There is no (159, 195, 8381556)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2521 731288 486173 562356 800608 664818 429583 652937 398647 518409 148443 455005 226706 851879 683866 880253 083688 324066 997641 825217 > 4195 [i]