Best Known (157, 193, s)-Nets in Base 4
(157, 193, 1539)-Net over F4 — Constructive and digital
Digital (157, 193, 1539)-net over F4, using
- 41 times duplication [i] based on 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
- trace code for nets [i] based on digital (28, 64, 513)-net over F64, using
(157, 193, 11303)-Net over F4 — Digital
Digital (157, 193, 11303)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4193, 11303, F4, 36) (dual of [11303, 11110, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, 16405, F4, 36) (dual of [16405, 16212, 37]-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(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(36) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(4193, 16405, F4, 36) (dual of [16405, 16212, 37]-code), using
(157, 193, 7185035)-Net in Base 4 — Upper bound on s
There is no (157, 193, 7185036)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 157 608056 286017 529328 088909 332740 131715 043005 506452 167357 717895 321848 946194 326399 641480 972849 759870 220026 461409 395640 > 4193 [i]