Best Known (215−41, 215, s)-Nets in Base 4
(215−41, 215, 1539)-Net over F4 — Constructive and digital
Digital (174, 215, 1539)-net over F4, using
- 4 times m-reduction [i] based on digital (174, 219, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 73, 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, 73, 513)-net over F64, using
(215−41, 215, 10294)-Net over F4 — Digital
Digital (174, 215, 10294)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4215, 10294, F4, 41) (dual of [10294, 10079, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4215, 16402, F4, 41) (dual of [16402, 16187, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- linear OA(4211, 16385, F4, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4197, 16385, F4, 37) (dual of [16385, 16188, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4215, 16402, F4, 41) (dual of [16402, 16187, 42]-code), using
(215−41, 215, 7659950)-Net in Base 4 — Upper bound on s
There is no (174, 215, 7659951)-net in base 4, because
- 1 times m-reduction [i] would yield (174, 214, 7659951)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 693 168213 984495 502444 300473 781704 959088 308586 709151 726988 828364 073182 056969 657312 240297 563910 230067 985055 916135 192656 926799 125666 > 4214 [i]