Best Known (113, 113+17, s)-Nets in Base 4
(113, 113+17, 131081)-Net over F4 — Constructive and digital
Digital (113, 130, 131081)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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 (104, 121, 131072)-net over F4, using
- net defined by OOA [i] based on linear OOA(4121, 131072, F4, 17, 17) (dual of [(131072, 17), 2228103, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using
- net defined by OOA [i] based on linear OOA(4121, 131072, F4, 17, 17) (dual of [(131072, 17), 2228103, 18]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(113, 113+17, 524313)-Net over F4 — Digital
Digital (113, 130, 524313)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4130, 524313, F4, 2, 17) (dual of [(524313, 2), 1048496, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4130, 1048626, F4, 17) (dual of [1048626, 1048496, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(481, 1048577, F4, 11) (dual of [1048577, 1048496, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(49, 49, F4, 5) (dual of [49, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(4130, 1048626, F4, 17) (dual of [1048626, 1048496, 18]-code), using
(113, 113+17, large)-Net in Base 4 — Upper bound on s
There is no (113, 130, large)-net in base 4, because
- 15 times m-reduction [i] would yield (113, 115, large)-net in base 4, but