Best Known (29−15, 29, s)-Nets in Base 25
(29−15, 29, 126)-Net over F25 — Constructive and digital
Digital (14, 29, 126)-net over F25, using
- t-expansion [i] based on digital (10, 29, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
(29−15, 29, 303)-Net over F25 — Digital
Digital (14, 29, 303)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2529, 303, F25, 2, 15) (dual of [(303, 2), 577, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2529, 314, F25, 2, 15) (dual of [(314, 2), 599, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2529, 628, F25, 15) (dual of [628, 599, 16]-code), using
- construction XX applied to C1 = C([623,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([623,13]) [i] based on
- linear OA(2527, 624, F25, 14) (dual of [624, 597, 15]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2527, 624, F25, 14) (dual of [624, 597, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2529, 624, F25, 15) (dual of [624, 595, 16]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2525, 624, F25, 13) (dual of [624, 599, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [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([623,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([623,13]) [i] based on
- OOA 2-folding [i] based on linear OA(2529, 628, F25, 15) (dual of [628, 599, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(2529, 314, F25, 2, 15) (dual of [(314, 2), 599, 16]-NRT-code), using
(29−15, 29, 55009)-Net in Base 25 — Upper bound on s
There is no (14, 29, 55010)-net in base 25, because
- 1 times m-reduction [i] would yield (14, 28, 55010)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1387 783027 555848 040021 027950 840749 123089 > 2528 [i]