Best Known (29, 29+23, s)-Nets in Base 25
(29, 29+23, 208)-Net over F25 — Constructive and digital
Digital (29, 52, 208)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 20, 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 (9, 32, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25 (see above)
- digital (9, 20, 104)-net over F25, using
(29, 29+23, 778)-Net over F25 — Digital
Digital (29, 52, 778)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2552, 778, F25, 23) (dual of [778, 726, 24]-code), using
- 143 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 4 times 0, 1, 15 times 0, 1, 40 times 0, 1, 78 times 0) [i] based on linear OA(2545, 628, F25, 23) (dual of [628, 583, 24]-code), using
- construction XX applied to C1 = C([623,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([623,21]) [i] based on
- linear OA(2543, 624, F25, 22) (dual of [624, 581, 23]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [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(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(2541, 624, F25, 21) (dual of [624, 583, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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([623,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([623,21]) [i] based on
- 143 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 4 times 0, 1, 15 times 0, 1, 40 times 0, 1, 78 times 0) [i] based on linear OA(2545, 628, F25, 23) (dual of [628, 583, 24]-code), using
(29, 29+23, 619665)-Net in Base 25 — Upper bound on s
There is no (29, 52, 619666)-net in base 25, because
- 1 times m-reduction [i] would yield (29, 51, 619666)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 197217 008487 466723 819747 447437 807414 644011 085618 221647 035161 699031 058385 > 2551 [i]