Best Known (122−19, 122, s)-Nets in Base 4
(122−19, 122, 7287)-Net over F4 — Constructive and digital
Digital (103, 122, 7287)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (94, 113, 7282)-net over F4, using
- net defined by OOA [i] based on linear OOA(4113, 7282, F4, 19, 19) (dual of [(7282, 19), 138245, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4113, 65539, F4, 19) (dual of [65539, 65426, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 65544, F4, 19) (dual of [65544, 65431, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4113, 65536, F4, 19) (dual of [65536, 65423, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4113, 65544, F4, 19) (dual of [65544, 65431, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4113, 65539, F4, 19) (dual of [65539, 65426, 20]-code), using
- net defined by OOA [i] based on linear OOA(4113, 7282, F4, 19, 19) (dual of [(7282, 19), 138245, 20]-NRT-code), using
- digital (0, 9, 5)-net over F4, using
(122−19, 122, 46127)-Net over F4 — Digital
Digital (103, 122, 46127)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4122, 46127, F4, 19) (dual of [46127, 46005, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4122, 65563, F4, 19) (dual of [65563, 65441, 20]-code), using
- construction XX applied to C1 = C([0,8]), C2 = C([1,9]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([0,9]) [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(4112, 65537, F4, 10) (dual of [65537, 65425, 11]-code), using the narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [1,9], and minimum distance d ≥ |{−9,−7,−5,…,9}|+1 = 11 (BCH-bound) [i]
- linear OA(4113, 65537, F4, 19) (dual of [65537, 65424, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(496, 65537, F4, 8) (dual of [65537, 65441, 9]-code), using the narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([0,8]), C2 = C([1,9]), C3 = C1 + C2 = C([1,8]), and C∩ = C1 ∩ C2 = C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4122, 65563, F4, 19) (dual of [65563, 65441, 20]-code), using
(122−19, 122, large)-Net in Base 4 — Upper bound on s
There is no (103, 122, large)-net in base 4, because
- 17 times m-reduction [i] would yield (103, 105, large)-net in base 4, but