Best Known (52−27, 52, s)-Nets in Base 49
(52−27, 52, 344)-Net over F49 — Constructive and digital
Digital (25, 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
(52−27, 52, 601)-Net over F49 — Digital
Digital (25, 52, 601)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4952, 601, F49, 2, 27) (dual of [(601, 2), 1150, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4952, 1202, F49, 27) (dual of [1202, 1150, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 1203, F49, 27) (dual of [1203, 1151, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4952, 1201, F49, 27) (dual of [1201, 1149, 28]-code), using an extension Ce(26) of the narrow-sense BCH-code C(I) with length 1200 | 492−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4950, 1201, F49, 26) (dual of [1201, 1151, 27]-code), using an extension Ce(25) of the narrow-sense BCH-code C(I) with length 1200 | 492−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4952, 1203, F49, 27) (dual of [1203, 1151, 28]-code), using
- OOA 2-folding [i] based on linear OA(4952, 1202, F49, 27) (dual of [1202, 1150, 28]-code), using
(52−27, 52, 504578)-Net in Base 49 — Upper bound on s
There is no (25, 52, 504579)-net in base 49, because
- 1 times m-reduction [i] would yield (25, 51, 504579)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 158 491204 778666 214647 020034 943855 812286 939665 707811 545022 364272 400953 949949 364898 040657 > 4951 [i]