Best Known (107−27, 107, s)-Nets in Base 5
(107−27, 107, 400)-Net over F5 — Constructive and digital
Digital (80, 107, 400)-net over F5, using
- 3 times m-reduction [i] based on digital (80, 110, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 55, 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, 55, 200)-net over F25, using
(107−27, 107, 2323)-Net over F5 — Digital
Digital (80, 107, 2323)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5107, 2323, F5, 27) (dual of [2323, 2216, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5107, 3131, F5, 27) (dual of [3131, 3024, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5106, 3130, F5, 27) (dual of [3130, 3024, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(5106, 3125, F5, 27) (dual of [3125, 3019, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 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(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5106, 3130, F5, 27) (dual of [3130, 3024, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5107, 3131, F5, 27) (dual of [3131, 3024, 28]-code), using
(107−27, 107, 708980)-Net in Base 5 — Upper bound on s
There is no (80, 107, 708981)-net in base 5, because
- 1 times m-reduction [i] would yield (80, 106, 708981)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 123 260233 831215 366616 877920 111475 580585 082811 597827 883973 374781 481085 722725 > 5106 [i]