Best Known (20, 20+15, s)-Nets in Base 49
(20, 20+15, 345)-Net over F49 — Constructive and digital
Digital (20, 35, 345)-net over F49, using
- 491 times duplication [i] based on digital (19, 34, 345)-net over F49, using
- net defined by OOA [i] based on linear OOA(4934, 345, F49, 15, 15) (dual of [(345, 15), 5141, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4934, 2416, F49, 15) (dual of [2416, 2382, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4934, 2419, F49, 15) (dual of [2419, 2385, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(4929, 2402, F49, 15) (dual of [2402, 2373, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(4917, 2402, F49, 9) (dual of [2402, 2385, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(495, 17, F49, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4934, 2419, F49, 15) (dual of [2419, 2385, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4934, 2416, F49, 15) (dual of [2416, 2382, 16]-code), using
- net defined by OOA [i] based on linear OOA(4934, 345, F49, 15, 15) (dual of [(345, 15), 5141, 16]-NRT-code), using
(20, 20+15, 2580)-Net over F49 — Digital
Digital (20, 35, 2580)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4935, 2580, F49, 15) (dual of [2580, 2545, 16]-code), using
- 171 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 8 times 0, 1, 35 times 0, 1, 123 times 0) [i] based on linear OA(4929, 2403, F49, 15) (dual of [2403, 2374, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4929, 2401, F49, 15) (dual of [2401, 2372, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- 171 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 8 times 0, 1, 35 times 0, 1, 123 times 0) [i] based on linear OA(4929, 2403, F49, 15) (dual of [2403, 2374, 16]-code), using
(20, 20+15, large)-Net in Base 49 — Upper bound on s
There is no (20, 35, large)-net in base 49, because
- 13 times m-reduction [i] would yield (20, 22, large)-net in base 49, but