Best Known (27, 34, s)-Nets in Base 4
(27, 34, 1372)-Net over F4 — Constructive and digital
Digital (27, 34, 1372)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (24, 31, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(431, 1367, F4, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(431, 4102, F4, 7) (dual of [4102, 4071, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(431, 4102, F4, 7) (dual of [4102, 4071, 8]-code), using
- net defined by OOA [i] based on linear OOA(431, 1367, F4, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
- digital (0, 3, 5)-net over F4, using
(27, 34, 4149)-Net over F4 — Digital
Digital (27, 34, 4149)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(434, 4149, F4, 7) (dual of [4149, 4115, 8]-code), using
- 37 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 29 times 0) [i] based on linear OA(432, 4110, F4, 7) (dual of [4110, 4078, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(413, 14, F4, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,4)), using
- dual of repetition code with length 14 [i]
- linear OA(41, 14, F4, 1) (dual of [14, 13, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 37 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 29 times 0) [i] based on linear OA(432, 4110, F4, 7) (dual of [4110, 4078, 8]-code), using
(27, 34, 2540516)-Net in Base 4 — Upper bound on s
There is no (27, 34, 2540517)-net in base 4, because
- 1 times m-reduction [i] would yield (27, 33, 2540517)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 73 787000 372360 620948 > 433 [i]