Best Known (54−29, 54, s)-Nets in Base 25
(54−29, 54, 200)-Net over F25 — Constructive and digital
Digital (25, 54, 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
(54−29, 54, 262)-Net over F25 — Digital
Digital (25, 54, 262)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2554, 262, F25, 2, 29) (dual of [(262, 2), 470, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2554, 314, F25, 2, 29) (dual of [(314, 2), 574, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2554, 628, F25, 29) (dual of [628, 574, 30]-code), using
- construction XX applied to C1 = C([623,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([623,27]) [i] based on
- linear OA(2552, 624, F25, 28) (dual of [624, 572, 29]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,26}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2552, 624, F25, 28) (dual of [624, 572, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2554, 624, F25, 29) (dual of [624, 570, 30]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2550, 624, F25, 27) (dual of [624, 574, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [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([623,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([623,27]) [i] based on
- OOA 2-folding [i] based on linear OA(2554, 628, F25, 29) (dual of [628, 574, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2554, 314, F25, 2, 29) (dual of [(314, 2), 574, 30]-NRT-code), using
(54−29, 54, 49361)-Net in Base 25 — Upper bound on s
There is no (25, 54, 49362)-net in base 25, because
- 1 times m-reduction [i] would yield (25, 53, 49362)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 123 287739 685708 041393 442977 029041 422568 630534 702545 923288 175387 755032 211425 > 2553 [i]