Best Known (23, 23+23, s)-Nets in Base 49
(23, 23+23, 344)-Net over F49 — Constructive and digital
Digital (23, 46, 344)-net over F49, using
- t-expansion [i] based on digital (21, 46, 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
(23, 23+23, 895)-Net over F49 — Digital
Digital (23, 46, 895)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4946, 895, F49, 2, 23) (dual of [(895, 2), 1744, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4946, 1203, F49, 2, 23) (dual of [(1203, 2), 2360, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4946, 2406, F49, 23) (dual of [2406, 2360, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4946, 2407, F49, 23) (dual of [2407, 2361, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [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(4941, 2402, F49, 21) (dual of [2402, 2361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 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([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4946, 2407, F49, 23) (dual of [2407, 2361, 24]-code), using
- OOA 2-folding [i] based on linear OA(4946, 2406, F49, 23) (dual of [2406, 2360, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(4946, 1203, F49, 2, 23) (dual of [(1203, 2), 2360, 24]-NRT-code), using
(23, 23+23, 839863)-Net in Base 49 — Upper bound on s
There is no (23, 46, 839864)-net in base 49, because
- 1 times m-reduction [i] would yield (23, 45, 839864)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 11450 547780 719704 443415 774401 260215 954662 778288 386809 474580 699762 069850 013569 > 4945 [i]