Best Known (27−11, 27, s)-Nets in Base 4
(27−11, 27, 66)-Net over F4 — Constructive and digital
Digital (16, 27, 66)-net over F4, using
- 1 times m-reduction [i] based on digital (16, 28, 66)-net over F4, using
- trace code for nets [i] based on digital (2, 14, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- trace code for nets [i] based on digital (2, 14, 33)-net over F16, using
(27−11, 27, 71)-Net over F4 — Digital
Digital (16, 27, 71)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(427, 71, F4, 11) (dual of [71, 44, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(427, 74, F4, 11) (dual of [74, 47, 12]-code), using
- construction XX applied to C1 = C({0,1,2,3,5,6,31,47}), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C({0,1,2,3,5,6,7,31,47}) [i] based on
- linear OA(422, 63, F4, 9) (dual of [63, 41, 10]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,5,6,31,47}, and minimum distance d ≥ |{−2,−1,…,6}|+1 = 10 (BCH-bound) [i]
- linear OA(419, 63, F4, 9) (dual of [63, 44, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(425, 63, F4, 11) (dual of [63, 38, 12]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,5,6,7,31,47}, and minimum distance d ≥ |{−2,−1,…,8}|+1 = 12 (BCH-bound) [i]
- linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- construction XX applied to C1 = C({0,1,2,3,5,6,31,47}), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C({0,1,2,3,5,6,7,31,47}) [i] based on
- discarding factors / shortening the dual code based on linear OA(427, 74, F4, 11) (dual of [74, 47, 12]-code), using
(27−11, 27, 1169)-Net in Base 4 — Upper bound on s
There is no (16, 27, 1170)-net in base 4, because
- 1 times m-reduction [i] would yield (16, 26, 1170)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4509 623724 642388 > 426 [i]