Best Known (44−20, 44, s)-Nets in Base 25
(44−20, 44, 192)-Net over F25 — Constructive and digital
Digital (24, 44, 192)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (10, 30, 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
- digital (4, 14, 66)-net over F25, using
(44−20, 44, 663)-Net over F25 — Digital
Digital (24, 44, 663)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2544, 663, F25, 20) (dual of [663, 619, 21]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 6 times 0, 1, 20 times 0) [i] based on linear OA(2539, 628, F25, 20) (dual of [628, 589, 21]-code), using
- construction XX applied to C1 = C([623,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([623,18]) [i] based on
- linear OA(2537, 624, F25, 19) (dual of [624, 587, 20]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [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(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(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [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([623,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([623,18]) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 6 times 0, 1, 20 times 0) [i] based on linear OA(2539, 628, F25, 20) (dual of [628, 589, 21]-code), using
(44−20, 44, 267111)-Net in Base 25 — Upper bound on s
There is no (24, 44, 267112)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 32 311960 850362 191756 140902 407537 219241 952520 828787 083074 816641 > 2544 [i]