Best Known (23, 23+24, s)-Nets in Base 49
(23, 23+24, 344)-Net over F49 — Constructive and digital
Digital (23, 47, 344)-net over F49, using
- t-expansion [i] based on digital (21, 47, 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+24, 801)-Net over F49 — Digital
Digital (23, 47, 801)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4947, 801, F49, 3, 24) (dual of [(801, 3), 2356, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4947, 2403, F49, 24) (dual of [2403, 2356, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(4947, 2401, F49, 24) (dual of [2401, 2354, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(4947, 2403, F49, 24) (dual of [2403, 2356, 25]-code), using
(23, 23+24, 459250)-Net in Base 49 — Upper bound on s
There is no (23, 47, 459251)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 27 493147 489854 537888 934717 419333 243639 882826 935363 925936 086184 789071 722726 481601 > 4947 [i]