Best Known (88, 88+17, s)-Nets in Base 4
(88, 88+17, 8197)-Net over F4 — Constructive and digital
Digital (88, 105, 8197)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-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 (0, 8, 5)-net over F4, using
(88, 88+17, 32785)-Net over F4 — Digital
Digital (88, 105, 32785)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4105, 32785, F4, 2, 17) (dual of [(32785, 2), 65465, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4105, 65570, F4, 17) (dual of [65570, 65465, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(45, 31, F4, 3) (dual of [31, 26, 4]-code or 31-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(4105, 65570, F4, 17) (dual of [65570, 65465, 18]-code), using
(88, 88+17, large)-Net in Base 4 — Upper bound on s
There is no (88, 105, large)-net in base 4, because
- 15 times m-reduction [i] would yield (88, 90, large)-net in base 4, but