Best Known (77, 103, s)-Nets in Base 5
(77, 103, 400)-Net over F5 — Constructive and digital
Digital (77, 103, 400)-net over F5, using
- 1 times m-reduction [i] based on digital (77, 104, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 52, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 52, 200)-net over F25, using
(77, 103, 2274)-Net over F5 — Digital
Digital (77, 103, 2274)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5103, 2274, F5, 26) (dual of [2274, 2171, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(5103, 3133, F5, 26) (dual of [3133, 3030, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(596, 3125, F5, 24) (dual of [3125, 3029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(591, 3125, F5, 23) (dual of [3125, 3034, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5103, 3133, F5, 26) (dual of [3133, 3030, 27]-code), using
(77, 103, 489025)-Net in Base 5 — Upper bound on s
There is no (77, 103, 489026)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 986101 702535 633097 889594 231410 700451 742501 511574 288996 891740 499880 258825 > 5103 [i]