Best Known (28, 51, s)-Nets in Base 25
(28, 51, 200)-Net over F25 — Constructive and digital
Digital (28, 51, 200)-net over F25, using
- t-expansion [i] based on digital (25, 51, 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
(28, 51, 698)-Net over F25 — Digital
Digital (28, 51, 698)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2551, 698, F25, 23) (dual of [698, 647, 24]-code), using
- 64 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 4 times 0, 1, 15 times 0, 1, 40 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
- 64 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 4 times 0, 1, 15 times 0, 1, 40 times 0) [i] based on linear OA(2545, 628, F25, 23) (dual of [628, 583, 24]-code), using
(28, 51, 462456)-Net in Base 25 — Upper bound on s
There is no (28, 51, 462457)-net in base 25, because
- 1 times m-reduction [i] would yield (28, 50, 462457)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 7888 769352 909387 463135 133333 205180 498166 708940 446773 401189 332060 199305 > 2550 [i]