Best Known (29, 29+23, s)-Nets in Base 49
(29, 29+23, 344)-Net over F49 — Constructive and digital
Digital (29, 52, 344)-net over F49, using
- t-expansion [i] based on digital (21, 52, 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+23, 2292)-Net over F49 — Digital
Digital (29, 52, 2292)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4952, 2292, F49, 23) (dual of [2292, 2240, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 2425, F49, 23) (dual of [2425, 2373, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- linear OA(4945, 2402, F49, 23) (dual of [2402, 2357, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(497, 23, F49, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,49)), using
- discarding factors / shortening the dual code based on linear OA(497, 49, F49, 7) (dual of [49, 42, 8]-code or 49-arc in PG(6,49)), using
- Reed–Solomon code RS(42,49) [i]
- discarding factors / shortening the dual code based on linear OA(497, 49, F49, 7) (dual of [49, 42, 8]-code or 49-arc in PG(6,49)), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4952, 2425, F49, 23) (dual of [2425, 2373, 24]-code), using
(29, 29+23, 7016753)-Net in Base 49 — Upper bound on s
There is no (29, 52, 7016754)-net in base 49, because
- 1 times m-reduction [i] would yield (29, 51, 7016754)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 158 489397 000239 411941 771911 617104 432212 226809 026729 106759 294159 700253 407448 581956 766753 > 4951 [i]