Best Known (26−6, 26, s)-Nets in Base 4
(26−6, 26, 1368)-Net over F4 — Constructive and digital
Digital (20, 26, 1368)-net over F4, using
- net defined by OOA [i] based on linear OOA(426, 1368, F4, 6, 6) (dual of [(1368, 6), 8182, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(426, 4104, F4, 6) (dual of [4104, 4078, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(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(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- dual of repetition code with length 8 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 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(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(426, 4104, F4, 6) (dual of [4104, 4078, 7]-code), using
(26−6, 26, 4104)-Net over F4 — Digital
Digital (20, 26, 4104)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(426, 4104, F4, 6) (dual of [4104, 4078, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(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(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- dual of repetition code with length 8 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 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(5) ⊂ Ce(4) [i] based on
(26−6, 26, 100024)-Net in Base 4 — Upper bound on s
There is no (20, 26, 100025)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 4503 645981 987976 > 426 [i]