Best Known (148−26, 148, s)-Nets in Base 4
(148−26, 148, 1269)-Net over F4 — Constructive and digital
Digital (122, 148, 1269)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (108, 134, 1260)-net over F4, using
- net defined by OOA [i] based on linear OOA(4134, 1260, F4, 26, 26) (dual of [(1260, 26), 32626, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4134, 16380, F4, 26) (dual of [16380, 16246, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4134, 16380, F4, 26) (dual of [16380, 16246, 27]-code), using
- net defined by OOA [i] based on linear OOA(4134, 1260, F4, 26, 26) (dual of [(1260, 26), 32626, 27]-NRT-code), using
- digital (1, 14, 9)-net over F4, using
(148−26, 148, 15898)-Net over F4 — Digital
Digital (122, 148, 15898)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4148, 15898, F4, 26) (dual of [15898, 15750, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4148, 16420, F4, 26) (dual of [16420, 16272, 27]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(4141, 16385, F4, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4113, 16385, F4, 21) (dual of [16385, 16272, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4148, 16420, F4, 26) (dual of [16420, 16272, 27]-code), using
(148−26, 148, large)-Net in Base 4 — Upper bound on s
There is no (122, 148, large)-net in base 4, because
- 24 times m-reduction [i] would yield (122, 124, large)-net in base 4, but