Best Known (156, 156+36, s)-Nets in Base 4
(156, 156+36, 1539)-Net over F4 — Constructive and digital
Digital (156, 192, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 64, 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
(156, 156+36, 10850)-Net over F4 — Digital
Digital (156, 192, 10850)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4192, 10850, F4, 36) (dual of [10850, 10658, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4192, 16401, F4, 36) (dual of [16401, 16209, 37]-code), using
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ 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(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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(41, 16, F4, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(4192, 16401, F4, 36) (dual of [16401, 16209, 37]-code), using
(156, 156+36, 6652441)-Net in Base 4 — Upper bound on s
There is no (156, 192, 6652442)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 39 402007 199368 546937 129846 078118 128257 299325 694491 085317 496968 886664 911236 755980 013343 042084 292188 257518 443028 510836 > 4192 [i]