Best Known (74−34, 74, s)-Nets in Base 25
(74−34, 74, 252)-Net over F25 — Constructive and digital
Digital (40, 74, 252)-net over F25, using
- 6 times m-reduction [i] based on digital (40, 80, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- 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 (10, 50, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 30, 126)-net over F25, using
- (u, u+v)-construction [i] based on
(74−34, 74, 770)-Net over F25 — Digital
Digital (40, 74, 770)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2574, 770, F25, 34) (dual of [770, 696, 35]-code), using
- 132 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 36 times 0, 1, 55 times 0) [i] based on linear OA(2564, 628, F25, 34) (dual of [628, 564, 35]-code), using
- construction XX applied to C1 = C([623,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([623,32]) [i] based on
- 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(2562, 624, F25, 33) (dual of [624, 562, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2564, 624, F25, 34) (dual of [624, 560, 35]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [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(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([623,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([623,32]) [i] based on
- 132 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 36 times 0, 1, 55 times 0) [i] based on linear OA(2564, 628, F25, 34) (dual of [628, 564, 35]-code), using
(74−34, 74, 363809)-Net in Base 25 — Upper bound on s
There is no (40, 74, 363810)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 28 026448 200116 486746 628962 727568 244167 241156 870759 793522 997623 762740 068777 054111 482972 222331 114047 579185 > 2574 [i]