Best Known (63, 63+11, s)-Nets in Base 4
(63, 63+11, 52432)-Net over F4 — Constructive and digital
Digital (63, 74, 52432)-net over F4, using
- net defined by OOA [i] based on linear OOA(474, 52432, F4, 11, 11) (dual of [(52432, 11), 576678, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(474, 262161, F4, 11) (dual of [262161, 262087, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(474, 262164, F4, 11) (dual of [262164, 262090, 12]-code), using
- construction X 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(41, 19, F4, 1) (dual of [19, 18, 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 C([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(474, 262164, F4, 11) (dual of [262164, 262090, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(474, 262161, F4, 11) (dual of [262161, 262087, 12]-code), using
(63, 63+11, 131082)-Net over F4 — Digital
Digital (63, 74, 131082)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(474, 131082, F4, 2, 11) (dual of [(131082, 2), 262090, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(474, 262164, F4, 11) (dual of [262164, 262090, 12]-code), using
- construction X 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(41, 19, F4, 1) (dual of [19, 18, 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 C([0,5]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(474, 262164, F4, 11) (dual of [262164, 262090, 12]-code), using
(63, 63+11, large)-Net in Base 4 — Upper bound on s
There is no (63, 74, large)-net in base 4, because
- 9 times m-reduction [i] would yield (63, 65, large)-net in base 4, but