Best Known (104, 104+15, s)-Nets in Base 4
(104, 104+15, 149806)-Net over F4 — Constructive and digital
Digital (104, 119, 149806)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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 (96, 111, 149797)-net over F4, using
- net defined by OOA [i] based on linear OOA(4111, 149797, F4, 15, 15) (dual of [(149797, 15), 2246844, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4111, 1048580, F4, 15) (dual of [1048580, 1048469, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 1048586, F4, 15) (dual of [1048586, 1048475, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4111, 1048586, F4, 15) (dual of [1048586, 1048475, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4111, 1048580, F4, 15) (dual of [1048580, 1048469, 16]-code), using
- net defined by OOA [i] based on linear OOA(4111, 149797, F4, 15, 15) (dual of [(149797, 15), 2246844, 16]-NRT-code), using
- digital (1, 8, 9)-net over F4, using
(104, 104+15, 550972)-Net over F4 — Digital
Digital (104, 119, 550972)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4119, 550972, F4, 15) (dual of [550972, 550853, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4119, 1048586, F4, 15) (dual of [1048586, 1048467, 16]-code), using
- (u, u+v)-construction [i] based on
- linear OA(48, 10, F4, 7) (dual of [10, 2, 8]-code), using
- repeating each code word 2 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 2 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
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(48, 10, F4, 7) (dual of [10, 2, 8]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4119, 1048586, F4, 15) (dual of [1048586, 1048467, 16]-code), using
(104, 104+15, large)-Net in Base 4 — Upper bound on s
There is no (104, 119, large)-net in base 4, because
- 13 times m-reduction [i] would yield (104, 106, large)-net in base 4, but