Best Known (157, 157+37, s)-Nets in Base 4
(157, 157+37, 1126)-Net over F4 — Constructive and digital
Digital (157, 194, 1126)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (28, 46, 98)-net over F4, using
- trace code for nets [i] based on digital (5, 23, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 23, 49)-net over F16, using
- digital (111, 148, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 37, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 37, 257)-net over F256, using
- digital (28, 46, 98)-net over F4, using
(157, 157+37, 9656)-Net over F4 — Digital
Digital (157, 194, 9656)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4194, 9656, F4, 37) (dual of [9656, 9462, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 16403, F4, 37) (dual of [16403, 16209, 38]-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(43, 18, F4, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), 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(4194, 16403, F4, 37) (dual of [16403, 16209, 38]-code), using
(157, 157+37, 7185035)-Net in Base 4 — Upper bound on s
There is no (157, 194, 7185036)-net in base 4, because
- 1 times m-reduction [i] would yield (157, 193, 7185036)-net in base 4, but
- 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]