Best Known (29, 29+25, s)-Nets in Base 49
(29, 29+25, 344)-Net over F49 — Constructive and digital
Digital (29, 54, 344)-net over F49, using
- t-expansion [i] based on digital (21, 54, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(29, 29+25, 1531)-Net over F49 — Digital
Digital (29, 54, 1531)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4954, 1531, F49, 25) (dual of [1531, 1477, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4954, 2419, F49, 25) (dual of [2419, 2365, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4949, 2402, F49, 25) (dual of [2402, 2353, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4937, 2402, F49, 19) (dual of [2402, 2365, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4954, 2419, F49, 25) (dual of [2419, 2365, 26]-code), using
(29, 29+25, 3214786)-Net in Base 49 — Upper bound on s
There is no (29, 54, 3214787)-net in base 49, because
- 1 times m-reduction [i] would yield (29, 53, 3214787)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 380534 158607 889296 049277 141724 361245 766987 095225 459380 765904 405800 702488 565639 595226 604225 > 4953 [i]