Best Known (42−7, 42, s)-Nets in Base 4
(42−7, 42, 21851)-Net over F4 — Constructive and digital
Digital (35, 42, 21851)-net over F4, using
- net defined by OOA [i] based on linear OOA(442, 21851, F4, 7, 7) (dual of [(21851, 7), 152915, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(442, 65554, F4, 7) (dual of [65554, 65512, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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
- OOA 3-folding and stacking with additional row [i] based on linear OA(442, 65554, F4, 7) (dual of [65554, 65512, 8]-code), using
(42−7, 42, 65554)-Net over F4 — Digital
Digital (35, 42, 65554)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(442, 65554, F4, 7) (dual of [65554, 65512, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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
(42−7, 42, large)-Net in Base 4 — Upper bound on s
There is no (35, 42, large)-net in base 4, because
- 5 times m-reduction [i] would yield (35, 37, large)-net in base 4, but