Best Known (93, 93+30, s)-Nets in Base 5
(93, 93+30, 408)-Net over F5 — Constructive and digital
Digital (93, 123, 408)-net over F5, using
- 3 times m-reduction [i] based on digital (93, 126, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 63, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- trace code for nets [i] based on digital (30, 63, 204)-net over F25, using
(93, 93+30, 3117)-Net over F5 — Digital
Digital (93, 123, 3117)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5123, 3117, F5, 30) (dual of [3117, 2994, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 3132, F5, 30) (dual of [3132, 3009, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(5111, 3126, F5, 27) (dual of [3126, 3015, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(52, 6, F5, 2) (dual of [6, 4, 3]-code or 6-arc in PG(1,5)), using
- extended Reed–Solomon code RSe(4,5) [i]
- Hamming code H(2,5) [i]
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5123, 3132, F5, 30) (dual of [3132, 3009, 31]-code), using
(93, 93+30, 865476)-Net in Base 5 — Upper bound on s
There is no (93, 123, 865477)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 94 040328 612949 464119 936929 808403 062233 465821 408266 630959 621056 598114 344632 244055 211485 > 5123 [i]