Best Known (63−33, 63, s)-Nets in Base 25
(63−33, 63, 204)-Net over F25 — Constructive and digital
Digital (30, 63, 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
(63−33, 63, 315)-Net over F25 — Digital
Digital (30, 63, 315)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2563, 315, F25, 2, 33) (dual of [(315, 2), 567, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2563, 630, F25, 33) (dual of [630, 567, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2562, 625, F25, 33) (dual of [625, 563, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2558, 625, F25, 31) (dual of [625, 567, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(251, 5, F25, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(2563, 630, F25, 33) (dual of [630, 567, 34]-code), using
(63−33, 63, 74011)-Net in Base 25 — Upper bound on s
There is no (30, 63, 74012)-net in base 25, because
- 1 times m-reduction [i] would yield (30, 62, 74012)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 470 285070 140387 487924 045620 360769 407995 166751 363299 480313 146672 480406 144797 755336 564225 > 2562 [i]