Best Known (117−29, 117, s)-Nets in Base 5
(117−29, 117, 400)-Net over F5 — Constructive and digital
Digital (88, 117, 400)-net over F5, using
- 9 times m-reduction [i] based on digital (88, 126, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 63, 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, 63, 200)-net over F25, using
(117−29, 117, 2731)-Net over F5 — Digital
Digital (88, 117, 2731)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5117, 2731, F5, 29) (dual of [2731, 2614, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5117, 3136, F5, 29) (dual of [3136, 3019, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(5116, 3125, F5, 29) (dual of [3125, 3009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(51, 11, F5, 1) (dual of [11, 10, 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
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(5117, 3136, F5, 29) (dual of [3136, 3019, 30]-code), using
(117−29, 117, 935101)-Net in Base 5 — Upper bound on s
There is no (88, 117, 935102)-net in base 5, because
- 1 times m-reduction [i] would yield (88, 116, 935102)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1203 711331 852872 677345 547766 898283 082923 882665 167969 578019 630311 969533 809329 199425 > 5116 [i]