Best Known (180, 180+33, s)-Nets in Base 4
(180, 180+33, 4111)-Net over F4 — Constructive and digital
Digital (180, 213, 4111)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 20, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (160, 193, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(4193, 4096, F4, 33, 33) (dual of [(4096, 33), 134975, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- net defined by OOA [i] based on linear OOA(4193, 4096, F4, 33, 33) (dual of [(4096, 33), 134975, 34]-NRT-code), using
- digital (4, 20, 15)-net over F4, using
(180, 180+33, 54203)-Net over F4 — Digital
Digital (180, 213, 54203)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4213, 54203, F4, 33) (dual of [54203, 53990, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4213, 65559, F4, 33) (dual of [65559, 65346, 34]-code), using
- (u, u+v)-construction [i] based on
- linear OA(420, 22, F4, 16) (dual of [22, 2, 17]-code), using
- 3 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- repeating each code word 5 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- extended Reed–Solomon code RSe(2,4) [i]
- Simplex code S(2,4) [i]
- repeating each code word 5 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- 3 times truncation [i] based on linear OA(423, 25, F4, 19) (dual of [25, 2, 20]-code), using
- linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(420, 22, F4, 16) (dual of [22, 2, 17]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4213, 65559, F4, 33) (dual of [65559, 65346, 34]-code), using
(180, 180+33, large)-Net in Base 4 — Upper bound on s
There is no (180, 213, large)-net in base 4, because
- 31 times m-reduction [i] would yield (180, 182, large)-net in base 4, but