Best Known (16−7, 16, s)-Nets in Base 25
(16−7, 16, 650)-Net over F25 — Constructive and digital
Digital (9, 16, 650)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 26)-net over F25, using
- s-reduction based on digital (0, 0, s)-net over F25 with arbitrarily large s, using
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 1, 26)-net over F25, using
- s-reduction based on digital (0, 1, s)-net over F25 with arbitrarily large s, using
- digital (0, 1, 26)-net over F25 (see above)
- digital (0, 1, 26)-net over F25 (see above)
- digital (0, 1, 26)-net over F25 (see above)
- digital (0, 2, 26)-net over F25, using
- digital (0, 3, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- digital (0, 7, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25 (see above)
- digital (0, 0, 26)-net over F25, using
(16−7, 16, 776)-Net over F25 — Digital
Digital (9, 16, 776)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2516, 776, F25, 7) (dual of [776, 760, 8]-code), using
- 145 step Varšamov–Edel lengthening with (ri) = (2, 17 times 0, 1, 126 times 0) [i] based on linear OA(2513, 628, F25, 7) (dual of [628, 615, 8]-code), using
- construction XX applied to C1 = C([623,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([623,5]) [i] based on
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(259, 624, F25, 5) (dual of [624, 615, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([623,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([623,5]) [i] based on
- 145 step Varšamov–Edel lengthening with (ri) = (2, 17 times 0, 1, 126 times 0) [i] based on linear OA(2513, 628, F25, 7) (dual of [628, 615, 8]-code), using
(16−7, 16, 739387)-Net in Base 25 — Upper bound on s
There is no (9, 16, 739388)-net in base 25, because
- 1 times m-reduction [i] would yield (9, 15, 739388)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 931 325835 466511 563105 > 2515 [i]