Best Known (254−42, 254, s)-Nets in Base 4
(254−42, 254, 3122)-Net over F4 — Constructive and digital
Digital (212, 254, 3122)-net over F4, using
- net defined by OOA [i] based on linear OOA(4254, 3122, F4, 42, 42) (dual of [(3122, 42), 130870, 43]-NRT-code), using
- OA 21-folding and stacking [i] based on linear OA(4254, 65562, F4, 42) (dual of [65562, 65308, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4254, 65565, F4, 42) (dual of [65565, 65311, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(4254, 65565, F4, 42) (dual of [65565, 65311, 43]-code), using
- OA 21-folding and stacking [i] based on linear OA(4254, 65562, F4, 42) (dual of [65562, 65308, 43]-code), using
(254−42, 254, 33751)-Net over F4 — Digital
Digital (212, 254, 33751)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4254, 33751, F4, 42) (dual of [33751, 33497, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4254, 65565, F4, 42) (dual of [65565, 65311, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(4254, 65565, F4, 42) (dual of [65565, 65311, 43]-code), using
(254−42, 254, large)-Net in Base 4 — Upper bound on s
There is no (212, 254, large)-net in base 4, because
- 40 times m-reduction [i] would yield (212, 214, large)-net in base 4, but