Best Known (89, 118, s)-Nets in Base 5
(89, 118, 408)-Net over F5 — Constructive and digital
Digital (89, 118, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 59, 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
(89, 118, 2900)-Net over F5 — Digital
Digital (89, 118, 2900)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5118, 2900, F5, 29) (dual of [2900, 2782, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5118, 3138, F5, 29) (dual of [3138, 3020, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [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(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(51, 12, F5, 1) (dual of [12, 11, 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(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(5118, 3138, F5, 29) (dual of [3138, 3020, 30]-code), using
(89, 118, 1049024)-Net in Base 5 — Upper bound on s
There is no (89, 118, 1049025)-net in base 5, because
- 1 times m-reduction [i] would yield (89, 117, 1049025)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 6018 540975 048816 354616 454966 664440 230134 387313 068948 005501 503402 301082 219605 000681 > 5117 [i]