Best Known (27, 27+21, s)-Nets in Base 25
(27, 27+21, 200)-Net over F25 — Constructive and digital
Digital (27, 48, 200)-net over F25, using
- t-expansion [i] based on digital (25, 48, 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
(27, 27+21, 801)-Net over F25 — Digital
Digital (27, 48, 801)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2548, 801, F25, 21) (dual of [801, 753, 22]-code), using
- 166 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 5 times 0, 1, 18 times 0, 1, 46 times 0, 1, 91 times 0) [i] based on linear OA(2541, 628, F25, 21) (dual of [628, 587, 22]-code), using
- construction XX applied to C1 = C([623,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([623,19]) [i] based on
- linear OA(2539, 624, F25, 20) (dual of [624, 585, 21]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2539, 624, F25, 20) (dual of [624, 585, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2541, 624, F25, 21) (dual of [624, 583, 22]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2537, 624, F25, 19) (dual of [624, 587, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([623,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([623,19]) [i] based on
- 166 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 5 times 0, 1, 18 times 0, 1, 46 times 0, 1, 91 times 0) [i] based on linear OA(2541, 628, F25, 21) (dual of [628, 587, 22]-code), using
(27, 27+21, 701584)-Net in Base 25 — Upper bound on s
There is no (27, 48, 701585)-net in base 25, because
- 1 times m-reduction [i] would yield (27, 47, 701585)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 504873 726718 029716 315515 207052 671428 290305 937766 506163 642466 020145 > 2547 [i]