Best Known (142−17, 142, s)-Nets in Base 4
(142−17, 142, 524297)-Net over F4 — Constructive and digital
Digital (125, 142, 524297)-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 (116, 133, 524288)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 524288, F4, 17, 17) (dual of [(524288, 17), 8912763, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, 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(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
- net defined by OOA [i] based on linear OOA(4133, 524288, F4, 17, 17) (dual of [(524288, 17), 8912763, 18]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(142−17, 142, 2097178)-Net over F4 — Digital
Digital (125, 142, 2097178)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4142, 2097178, F4, 2, 17) (dual of [(2097178, 2), 4194214, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4142, 4194356, F4, 17) (dual of [4194356, 4194214, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(489, 4194305, F4, 11) (dual of [4194305, 4194216, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(4142, 4194356, F4, 17) (dual of [4194356, 4194214, 18]-code), using
(142−17, 142, large)-Net in Base 4 — Upper bound on s
There is no (125, 142, large)-net in base 4, because
- 15 times m-reduction [i] would yield (125, 127, large)-net in base 4, but