Best Known (129−19, 129, s)-Nets in Base 4
(129−19, 129, 29129)-Net over F4 — Constructive and digital
Digital (110, 129, 29129)-net over F4, using
- 41 times duplication [i] based on digital (109, 128, 29129)-net over F4, using
- net defined by OOA [i] based on linear OOA(4128, 29129, F4, 19, 19) (dual of [(29129, 19), 553323, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4128, 262162, F4, 19) (dual of [262162, 262034, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 262164, F4, 19) (dual of [262164, 262036, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4127, 262145, F4, 19) (dual of [262145, 262018, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 262164, F4, 19) (dual of [262164, 262036, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4128, 262162, F4, 19) (dual of [262162, 262034, 20]-code), using
- net defined by OOA [i] based on linear OOA(4128, 29129, F4, 19, 19) (dual of [(29129, 19), 553323, 20]-NRT-code), using
(129−19, 129, 131083)-Net over F4 — Digital
Digital (110, 129, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4129, 131083, F4, 2, 19) (dual of [(131083, 2), 262037, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4129, 262166, F4, 19) (dual of [262166, 262037, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4128, 262165, F4, 19) (dual of [262165, 262037, 20]-code), using
- construction X4 applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4127, 262145, F4, 19) (dual of [262145, 262018, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4128, 262165, F4, 19) (dual of [262165, 262037, 20]-code), using
- OOA 2-folding [i] based on linear OA(4129, 262166, F4, 19) (dual of [262166, 262037, 20]-code), using
(129−19, 129, large)-Net in Base 4 — Upper bound on s
There is no (110, 129, large)-net in base 4, because
- 17 times m-reduction [i] would yield (110, 112, large)-net in base 4, but