Best Known (260−40, 260, s)-Nets in Base 4
(260−40, 260, 3281)-Net over F4 — Constructive and digital
Digital (220, 260, 3281)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 20, 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 (200, 240, 3276)-net over F4, using
- net defined by OOA [i] based on linear OOA(4240, 3276, F4, 40, 40) (dual of [(3276, 40), 130800, 41]-NRT-code), using
- OA 20-folding and stacking [i] based on linear OA(4240, 65520, F4, 40) (dual of [65520, 65280, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 65536, F4, 40) (dual of [65536, 65296, 41]-code), using
- 1 times truncation [i] based on linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 65536, F4, 40) (dual of [65536, 65296, 41]-code), using
- OA 20-folding and stacking [i] based on linear OA(4240, 65520, F4, 40) (dual of [65520, 65280, 41]-code), using
- net defined by OOA [i] based on linear OOA(4240, 3276, F4, 40, 40) (dual of [(3276, 40), 130800, 41]-NRT-code), using
- digital (0, 20, 5)-net over F4, using
(260−40, 260, 63531)-Net over F4 — Digital
Digital (220, 260, 63531)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4260, 63531, F4, 40) (dual of [63531, 63271, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4260, 65557, F4, 40) (dual of [65557, 65297, 41]-code), using
- (u, u+v)-construction [i] based on
- linear OA(420, 21, F4, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,4)), using
- dual of repetition code with length 21 [i]
- linear OA(4240, 65536, F4, 40) (dual of [65536, 65296, 41]-code), using
- 1 times truncation [i] based on linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using
- linear OA(420, 21, F4, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4260, 65557, F4, 40) (dual of [65557, 65297, 41]-code), using
(260−40, 260, large)-Net in Base 4 — Upper bound on s
There is no (220, 260, large)-net in base 4, because
- 38 times m-reduction [i] would yield (220, 222, large)-net in base 4, but