Best Known (157−25, 157, s)-Nets in Base 4
(157−25, 157, 5466)-Net over F4 — Constructive and digital
Digital (132, 157, 5466)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (120, 145, 5461)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 5461, F4, 25, 25) (dual of [(5461, 25), 136380, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4145, 65533, F4, 25) (dual of [65533, 65388, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4145, 65533, F4, 25) (dual of [65533, 65388, 26]-code), using
- net defined by OOA [i] based on linear OOA(4145, 5461, F4, 25, 25) (dual of [(5461, 25), 136380, 26]-NRT-code), using
- digital (0, 12, 5)-net over F4, using
(157−25, 157, 38077)-Net over F4 — Digital
Digital (132, 157, 38077)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4157, 38077, F4, 25) (dual of [38077, 37920, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4157, 65550, F4, 25) (dual of [65550, 65393, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([1,12]) [i] based on
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4144, 65537, F4, 12) (dual of [65537, 65393, 13]-code), using the narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- construction X applied to C([0,12]) ⊂ C([1,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4157, 65550, F4, 25) (dual of [65550, 65393, 26]-code), using
(157−25, 157, large)-Net in Base 4 — Upper bound on s
There is no (132, 157, large)-net in base 4, because
- 23 times m-reduction [i] would yield (132, 134, large)-net in base 4, but