Best Known (77−12, 77, s)-Nets in Base 4
(77−12, 77, 10927)-Net over F4 — Constructive and digital
Digital (65, 77, 10927)-net over F4, using
- net defined by OOA [i] based on linear OOA(477, 10927, F4, 12, 12) (dual of [(10927, 12), 131047, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(477, 65562, F4, 12) (dual of [65562, 65485, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 65564, F4, 12) (dual of [65564, 65487, 13]-code), using
- 1 times truncation [i] based on linear OA(478, 65565, F4, 13) (dual of [65565, 65487, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 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(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(12) ⊂ Ce(8) [i] based on
- 1 times truncation [i] based on linear OA(478, 65565, F4, 13) (dual of [65565, 65487, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 65564, F4, 12) (dual of [65564, 65487, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(477, 65562, F4, 12) (dual of [65562, 65485, 13]-code), using
(77−12, 77, 56814)-Net over F4 — Digital
Digital (65, 77, 56814)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(477, 56814, F4, 12) (dual of [56814, 56737, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 65564, F4, 12) (dual of [65564, 65487, 13]-code), using
- 1 times truncation [i] based on linear OA(478, 65565, F4, 13) (dual of [65565, 65487, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 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(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(12) ⊂ Ce(8) [i] based on
- 1 times truncation [i] based on linear OA(478, 65565, F4, 13) (dual of [65565, 65487, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 65564, F4, 12) (dual of [65564, 65487, 13]-code), using
(77−12, 77, large)-Net in Base 4 — Upper bound on s
There is no (65, 77, large)-net in base 4, because
- 10 times m-reduction [i] would yield (65, 67, large)-net in base 4, but