Best Known (253−39, 253, s)-Nets in Base 4
(253−39, 253, 3458)-Net over F4 — Constructive and digital
Digital (214, 253, 3458)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 20, 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 (194, 233, 3449)-net over F4, using
- net defined by OOA [i] based on linear OOA(4233, 3449, F4, 39, 39) (dual of [(3449, 39), 134278, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4233, 65532, F4, 39) (dual of [65532, 65299, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using
- an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- discarding factors / shortening the dual code based on linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4233, 65532, F4, 39) (dual of [65532, 65299, 40]-code), using
- net defined by OOA [i] based on linear OOA(4233, 3449, F4, 39, 39) (dual of [(3449, 39), 134278, 40]-NRT-code), using
- digital (1, 20, 9)-net over F4, using
(253−39, 253, 61531)-Net over F4 — Digital
Digital (214, 253, 61531)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4253, 61531, F4, 39) (dual of [61531, 61278, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4253, 65609, F4, 39) (dual of [65609, 65356, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(29) [i] based on
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(38) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4253, 65609, F4, 39) (dual of [65609, 65356, 40]-code), using
(253−39, 253, large)-Net in Base 4 — Upper bound on s
There is no (214, 253, large)-net in base 4, because
- 37 times m-reduction [i] would yield (214, 216, large)-net in base 4, but