Best Known (79−10, 79, s)-Nets in Base 4
(79−10, 79, 838863)-Net over F4 — Constructive and digital
Digital (69, 79, 838863)-net over F4, using
- 41 times duplication [i] based on digital (68, 78, 838863)-net over F4, using
- net defined by OOA [i] based on linear OOA(478, 838863, F4, 10, 10) (dual of [(838863, 10), 8388552, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(478, 4194315, F4, 10) (dual of [4194315, 4194237, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 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(40, 11, F4, 0) (dual of [11, 11, 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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(478, 4194315, F4, 10) (dual of [4194315, 4194237, 11]-code), using
- net defined by OOA [i] based on linear OOA(478, 838863, F4, 10, 10) (dual of [(838863, 10), 8388552, 11]-NRT-code), using
(79−10, 79, 2097158)-Net over F4 — Digital
Digital (69, 79, 2097158)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(479, 2097158, F4, 2, 10) (dual of [(2097158, 2), 4194237, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(479, 4194316, F4, 10) (dual of [4194316, 4194237, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(479, 4194317, F4, 10) (dual of [4194317, 4194238, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 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(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(479, 4194317, F4, 10) (dual of [4194317, 4194238, 11]-code), using
- OOA 2-folding [i] based on linear OA(479, 4194316, F4, 10) (dual of [4194316, 4194237, 11]-code), using
(79−10, 79, large)-Net in Base 4 — Upper bound on s
There is no (69, 79, large)-net in base 4, because
- 8 times m-reduction [i] would yield (69, 71, large)-net in base 4, but