Best Known (66, 77, s)-Nets in Base 4
(66, 77, 52433)-Net over F4 — Constructive and digital
Digital (66, 77, 52433)-net over F4, using
- 42 times duplication [i] based on digital (64, 75, 52433)-net over F4, using
- net defined by OOA [i] based on linear OOA(475, 52433, F4, 11, 11) (dual of [(52433, 11), 576688, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(475, 262166, F4, 11) (dual of [262166, 262091, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(474, 262165, F4, 11) (dual of [262165, 262091, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(473, 262145, F4, 11) (dual of [262145, 262072, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(455, 262145, F4, 9) (dual of [262145, 262090, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 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 C([0,5]) ⊂ C([0,4]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(474, 262165, F4, 11) (dual of [262165, 262091, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(475, 262166, F4, 11) (dual of [262166, 262091, 12]-code), using
- net defined by OOA [i] based on linear OOA(475, 52433, F4, 11, 11) (dual of [(52433, 11), 576688, 12]-NRT-code), using
(66, 77, 167755)-Net over F4 — Digital
Digital (66, 77, 167755)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(477, 167755, F4, 11) (dual of [167755, 167678, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 262150, F4, 11) (dual of [262150, 262073, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([1,5]) [i] based on
- linear OA(473, 262145, F4, 11) (dual of [262145, 262072, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(472, 262145, F4, 6) (dual of [262145, 262073, 7]-code), using the narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [1,5], and minimum distance d ≥ |{−5,−3,−1,…,5}|+1 = 7 (BCH-bound) [i]
- linear OA(44, 5, F4, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,4)), using
- dual of repetition code with length 5 [i]
- construction X applied to C([0,5]) ⊂ C([1,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(477, 262150, F4, 11) (dual of [262150, 262073, 12]-code), using
(66, 77, large)-Net in Base 4 — Upper bound on s
There is no (66, 77, large)-net in base 4, because
- 9 times m-reduction [i] would yield (66, 68, large)-net in base 4, but