Best Known (97−22, 97, s)-Nets in Base 5
(97−22, 97, 400)-Net over F5 — Constructive and digital
Digital (75, 97, 400)-net over F5, using
- 3 times m-reduction [i] based on digital (75, 100, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 50, 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, 50, 200)-net over F25, using
(97−22, 97, 3738)-Net over F5 — Digital
Digital (75, 97, 3738)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(597, 3738, F5, 22) (dual of [3738, 3641, 23]-code), using
- 597 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0, 1, 102 times 0, 1, 152 times 0, 1, 207 times 0) [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- 597 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0, 1, 102 times 0, 1, 152 times 0, 1, 207 times 0) [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
(97−22, 97, 1788944)-Net in Base 5 — Upper bound on s
There is no (75, 97, 1788945)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 63 109066 691731 352289 138630 880765 547815 498900 043188 780973 890538 724893 > 597 [i]