Best Known (29, 29+33, s)-Nets in Base 25
(29, 29+33, 200)-Net over F25 — Constructive and digital
Digital (29, 62, 200)-net over F25, using
- t-expansion [i] based on digital (25, 62, 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
(29, 29+33, 299)-Net over F25 — Digital
Digital (29, 62, 299)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2562, 299, F25, 2, 33) (dual of [(299, 2), 536, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2562, 314, F25, 2, 33) (dual of [(314, 2), 566, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2562, 628, F25, 33) (dual of [628, 566, 34]-code), using
- construction XX applied to C1 = C([623,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([623,31]) [i] based on
- linear OA(2560, 624, F25, 32) (dual of [624, 564, 33]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2560, 624, F25, 32) (dual of [624, 564, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2562, 624, F25, 33) (dual of [624, 562, 34]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2558, 624, F25, 31) (dual of [624, 566, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([623,31]) [i] based on
- OOA 2-folding [i] based on linear OA(2562, 628, F25, 33) (dual of [628, 566, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2562, 314, F25, 2, 33) (dual of [(314, 2), 566, 34]-NRT-code), using
(29, 29+33, 60522)-Net in Base 25 — Upper bound on s
There is no (29, 62, 60523)-net in base 25, because
- 1 times m-reduction [i] would yield (29, 61, 60523)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 18 811566 218913 991830 294663 655146 671809 088256 935783 406806 367041 671429 254218 427089 296513 > 2561 [i]