Best Known (79, 79+25, s)-Nets in Base 3
(79, 79+25, 400)-Net over F3 — Constructive and digital
Digital (79, 104, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 26, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(79, 79+25, 626)-Net over F3 — Digital
Digital (79, 104, 626)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3104, 626, F3, 25) (dual of [626, 522, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3104, 754, F3, 25) (dual of [754, 650, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(379, 729, F3, 20) (dual of [729, 650, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(37, 25, F3, 4) (dual of [25, 18, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3104, 754, F3, 25) (dual of [754, 650, 26]-code), using
(79, 79+25, 32920)-Net in Base 3 — Upper bound on s
There is no (79, 104, 32921)-net in base 3, because
- 1 times m-reduction [i] would yield (79, 103, 32921)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13 916150 708220 245274 782998 848971 286091 489875 407409 > 3103 [i]