Best Known (89, 89+17, s)-Nets in Base 4
(89, 89+17, 8201)-Net over F4 — Constructive and digital
Digital (89, 106, 8201)-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 (80, 97, 8192)-net over F4, using
- net defined by OOA [i] based on linear OOA(497, 8192, F4, 17, 17) (dual of [(8192, 17), 139167, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, 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(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using
- net defined by OOA [i] based on linear OOA(497, 8192, F4, 17, 17) (dual of [(8192, 17), 139167, 18]-NRT-code), using
- digital (1, 9, 9)-net over F4, using
(89, 89+17, 35069)-Net over F4 — Digital
Digital (89, 106, 35069)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4106, 35069, F4, 17) (dual of [35069, 34963, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4106, 65573, F4, 17) (dual of [65573, 65467, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(465, 65537, F4, 11) (dual of [65537, 65472, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(49, 36, F4, 5) (dual of [36, 27, 6]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4106, 65573, F4, 17) (dual of [65573, 65467, 18]-code), using
(89, 89+17, large)-Net in Base 4 — Upper bound on s
There is no (89, 106, large)-net in base 4, because
- 15 times m-reduction [i] would yield (89, 91, large)-net in base 4, but