Best Known (228, 258, s)-Nets in Base 4
(228, 258, 279626)-Net over F4 — Constructive and digital
Digital (228, 258, 279626)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 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 (213, 243, 279621)-net over F4, using
- net defined by OOA [i] based on linear OOA(4243, 279621, F4, 30, 30) (dual of [(279621, 30), 8388387, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4243, 4194315, F4, 30) (dual of [4194315, 4194072, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 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(29) ⊂ Ce(28) [i] based on
- OA 15-folding and stacking [i] based on linear OA(4243, 4194315, F4, 30) (dual of [4194315, 4194072, 31]-code), using
- net defined by OOA [i] based on linear OOA(4243, 279621, F4, 30, 30) (dual of [(279621, 30), 8388387, 31]-NRT-code), using
- digital (0, 15, 5)-net over F4, using
(228, 258, 1860806)-Net over F4 — Digital
Digital (228, 258, 1860806)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4258, 1860806, F4, 2, 30) (dual of [(1860806, 2), 3721354, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4258, 2097187, F4, 2, 30) (dual of [(2097187, 2), 4194116, 31]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4256, 2097186, F4, 2, 30) (dual of [(2097186, 2), 4194116, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4256, 4194372, F4, 30) (dual of [4194372, 4194116, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(413, 68, F4, 6) (dual of [68, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 70, F4, 6) (dual of [70, 57, 7]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(4256, 4194372, F4, 30) (dual of [4194372, 4194116, 31]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4256, 2097186, F4, 2, 30) (dual of [(2097186, 2), 4194116, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4258, 2097187, F4, 2, 30) (dual of [(2097187, 2), 4194116, 31]-NRT-code), using
(228, 258, large)-Net in Base 4 — Upper bound on s
There is no (228, 258, large)-net in base 4, because
- 28 times m-reduction [i] would yield (228, 230, large)-net in base 4, but