Best Known (42−14, 42, s)-Nets in Base 49
(42−14, 42, 16808)-Net over F49 — Constructive and digital
Digital (28, 42, 16808)-net over F49, using
- 491 times duplication [i] based on digital (27, 41, 16808)-net over F49, using
- net defined by OOA [i] based on linear OOA(4941, 16808, F49, 14, 14) (dual of [(16808, 14), 235271, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4941, 117656, F49, 14) (dual of [117656, 117615, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- OA 7-folding and stacking [i] based on linear OA(4941, 117656, F49, 14) (dual of [117656, 117615, 15]-code), using
- net defined by OOA [i] based on linear OOA(4941, 16808, F49, 14, 14) (dual of [(16808, 14), 235271, 15]-NRT-code), using
(42−14, 42, 65603)-Net over F49 — Digital
Digital (28, 42, 65603)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4942, 65603, F49, 14) (dual of [65603, 65561, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4942, 117660, F49, 14) (dual of [117660, 117618, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4931, 117649, F49, 11) (dual of [117649, 117618, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(492, 11, F49, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4942, 117660, F49, 14) (dual of [117660, 117618, 15]-code), using
(42−14, 42, large)-Net in Base 49 — Upper bound on s
There is no (28, 42, large)-net in base 49, because
- 12 times m-reduction [i] would yield (28, 30, large)-net in base 49, but