Best Known (91−15, 91, s)-Nets in Base 4
(91−15, 91, 9364)-Net over F4 — Constructive and digital
Digital (76, 91, 9364)-net over F4, using
- 41 times duplication [i] based on digital (75, 90, 9364)-net over F4, using
- net defined by OOA [i] based on linear OOA(490, 9364, F4, 15, 15) (dual of [(9364, 15), 140370, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(490, 65549, F4, 15) (dual of [65549, 65459, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(490, 65553, F4, 15) (dual of [65553, 65463, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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 X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(490, 65553, F4, 15) (dual of [65553, 65463, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(490, 65549, F4, 15) (dual of [65549, 65459, 16]-code), using
- net defined by OOA [i] based on linear OOA(490, 9364, F4, 15, 15) (dual of [(9364, 15), 140370, 16]-NRT-code), using
(91−15, 91, 32777)-Net over F4 — Digital
Digital (76, 91, 32777)-net over F4, using
- 41 times duplication [i] based on digital (75, 90, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(490, 32777, F4, 2, 15) (dual of [(32777, 2), 65464, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(490, 65554, F4, 15) (dual of [65554, 65464, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(490, 65554, F4, 15) (dual of [65554, 65464, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(490, 32777, F4, 2, 15) (dual of [(32777, 2), 65464, 16]-NRT-code), using
(91−15, 91, large)-Net in Base 4 — Upper bound on s
There is no (76, 91, large)-net in base 4, because
- 13 times m-reduction [i] would yield (76, 78, large)-net in base 4, but