Best Known (30−13, 30, s)-Nets in Base 49
(30−13, 30, 403)-Net over F49 — Constructive and digital
Digital (17, 30, 403)-net over F49, using
- net defined by OOA [i] based on linear OOA(4930, 403, F49, 13, 13) (dual of [(403, 13), 5209, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4930, 2419, F49, 13) (dual of [2419, 2389, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- linear OA(4925, 2402, F49, 13) (dual of [2402, 2377, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4913, 2402, F49, 7) (dual of [2402, 2389, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,6]) ⊂ C([0,3]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(4930, 2419, F49, 13) (dual of [2419, 2389, 14]-code), using
(30−13, 30, 2537)-Net over F49 — Digital
Digital (17, 30, 2537)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4930, 2537, F49, 13) (dual of [2537, 2507, 14]-code), using
- 129 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 96 times 0) [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- 129 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 96 times 0) [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
(30−13, 30, large)-Net in Base 49 — Upper bound on s
There is no (17, 30, large)-net in base 49, because
- 11 times m-reduction [i] would yield (17, 19, large)-net in base 49, but