Best Known (229−35, 229, s)-Nets in Base 4
(229−35, 229, 3869)-Net over F4 — Constructive and digital
Digital (194, 229, 3869)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 20, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (174, 209, 3855)-net over F4, using
- net defined by OOA [i] based on linear OOA(4209, 3855, F4, 35, 35) (dual of [(3855, 35), 134716, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 17-folding and stacking with additional row [i] based on linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using
- net defined by OOA [i] based on linear OOA(4209, 3855, F4, 35, 35) (dual of [(3855, 35), 134716, 36]-NRT-code), using
- digital (3, 20, 14)-net over F4, using
(229−35, 229, 63349)-Net over F4 — Digital
Digital (194, 229, 63349)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4229, 63349, F4, 35) (dual of [63349, 63120, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4229, 65609, F4, 35) (dual of [65609, 65380, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(420, 73, F4, 8) (dual of [73, 53, 9]-code), using
- construction XX applied to C1 = C({0,1,2,3,31,47}), C2 = C([0,5]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C({0,1,2,3,5,31,47}) [i] based on
- linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,31,47}, and minimum distance d ≥ |{−2,−1,…,4}|+1 = 8 (BCH-bound) [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(419, 63, F4, 8) (dual of [63, 44, 9]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,5,31,47}, and minimum distance d ≥ |{−2,−1,…,5}|+1 = 9 (BCH-bound) [i]
- linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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
- linear OA(40, 3, F4, 0) (dual of [3, 3, 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 XX applied to C1 = C({0,1,2,3,31,47}), C2 = C([0,5]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C({0,1,2,3,5,31,47}) [i] based on
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4229, 65609, F4, 35) (dual of [65609, 65380, 36]-code), using
(229−35, 229, large)-Net in Base 4 — Upper bound on s
There is no (194, 229, large)-net in base 4, because
- 33 times m-reduction [i] would yield (194, 196, large)-net in base 4, but