Best Known (55−24, 55, s)-Nets in Base 25
(55−24, 55, 230)-Net over F25 — Constructive and digital
Digital (31, 55, 230)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 21, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (10, 34, 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 (9, 21, 104)-net over F25, using
(55−24, 55, 878)-Net over F25 — Digital
Digital (31, 55, 878)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2555, 878, F25, 24) (dual of [878, 823, 25]-code), using
- 242 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 14 times 0, 1, 38 times 0, 1, 74 times 0, 1, 106 times 0) [i] based on linear OA(2547, 628, F25, 24) (dual of [628, 581, 25]-code), using
- construction XX applied to C1 = C([623,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([623,22]) [i] based on
- linear OA(2545, 624, F25, 23) (dual of [624, 579, 24]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2545, 624, F25, 23) (dual of [624, 579, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2547, 624, F25, 24) (dual of [624, 577, 25]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2543, 624, F25, 22) (dual of [624, 581, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [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([623,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([623,22]) [i] based on
- 242 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 14 times 0, 1, 38 times 0, 1, 74 times 0, 1, 106 times 0) [i] based on linear OA(2547, 628, F25, 24) (dual of [628, 581, 25]-code), using
(55−24, 55, 562821)-Net in Base 25 — Upper bound on s
There is no (31, 55, 562822)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 77038 226319 186768 302456 638304 811110 476693 551160 361488 443821 462695 043895 952705 > 2555 [i]