Best Known (30, 30+15, s)-Nets in Base 49
(30, 30+15, 16808)-Net over F49 — Constructive and digital
Digital (30, 45, 16808)-net over F49, using
- 491 times duplication [i] based on digital (29, 44, 16808)-net over F49, using
- net defined by OOA [i] based on linear OOA(4944, 16808, F49, 15, 15) (dual of [(16808, 15), 252076, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4944, 117657, F49, 15) (dual of [117657, 117613, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(4943, 117650, F49, 15) (dual of [117650, 117607, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(4944, 117657, F49, 15) (dual of [117657, 117613, 16]-code), using
- net defined by OOA [i] based on linear OOA(4944, 16808, F49, 15, 15) (dual of [(16808, 15), 252076, 16]-NRT-code), using
(30, 30+15, 62057)-Net over F49 — Digital
Digital (30, 45, 62057)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4945, 62057, F49, 15) (dual of [62057, 62012, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4945, 117660, F49, 15) (dual of [117660, 117615, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(4943, 117649, F49, 15) (dual of [117649, 117606, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(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(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(4945, 117660, F49, 15) (dual of [117660, 117615, 16]-code), using
(30, 30+15, large)-Net in Base 49 — Upper bound on s
There is no (30, 45, large)-net in base 49, because
- 13 times m-reduction [i] would yield (30, 32, large)-net in base 49, but