Best Known (69−8, 69, s)-Nets in Base 4
(69−8, 69, 1048582)-Net over F4 — Constructive and digital
Digital (61, 69, 1048582)-net over F4, using
- 41 times duplication [i] based on digital (60, 68, 1048582)-net over F4, using
- net defined by OOA [i] based on linear OOA(468, 1048582, F4, 8, 8) (dual of [(1048582, 8), 8388588, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(468, 4194328, F4, 8) (dual of [4194328, 4194260, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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(8) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(468, 4194328, F4, 8) (dual of [4194328, 4194260, 9]-code), using
- net defined by OOA [i] based on linear OOA(468, 1048582, F4, 8, 8) (dual of [(1048582, 8), 8388588, 9]-NRT-code), using
(69−8, 69, 4194330)-Net over F4 — Digital
Digital (61, 69, 4194330)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(469, 4194330, F4, 8) (dual of [4194330, 4194261, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(468, 4194328, F4, 8) (dual of [4194328, 4194260, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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(8) ⊂ Ce(5) [i] based on
- linear OA(468, 4194329, F4, 7) (dual of [4194329, 4194261, 8]-code), using Gilbert–Varšamov bound and bm = 468 > Vbs−1(k−1) = 5512 746519 677817 912237 124646 478493 830739 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(468, 4194328, F4, 8) (dual of [4194328, 4194260, 9]-code), using
- construction X with Varšamov bound [i] based on
(69−8, 69, large)-Net in Base 4 — Upper bound on s
There is no (61, 69, large)-net in base 4, because
- 6 times m-reduction [i] would yield (61, 63, large)-net in base 4, but