Best Known (25, 25+18, s)-Nets in Base 25
(25, 25+18, 200)-Net over F25 — Constructive and digital
Digital (25, 43, 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
(25, 25+18, 1037)-Net over F25 — Digital
Digital (25, 43, 1037)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2543, 1037, F25, 18) (dual of [1037, 994, 19]-code), using
- 401 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 9 times 0, 1, 27 times 0, 1, 65 times 0, 1, 122 times 0, 1, 170 times 0) [i] based on linear OA(2535, 628, F25, 18) (dual of [628, 593, 19]-code), using
- construction XX applied to C1 = C([623,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([623,16]) [i] based on
- linear OA(2533, 624, F25, 17) (dual of [624, 591, 18]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2533, 624, F25, 17) (dual of [624, 591, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2531, 624, F25, 16) (dual of [624, 593, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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([623,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([623,16]) [i] based on
- 401 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 9 times 0, 1, 27 times 0, 1, 65 times 0, 1, 122 times 0, 1, 170 times 0) [i] based on linear OA(2535, 628, F25, 18) (dual of [628, 593, 19]-code), using
(25, 25+18, 825248)-Net in Base 25 — Upper bound on s
There is no (25, 43, 825249)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 1 292480 277032 202250 686423 352864 631506 758353 427904 369568 237785 > 2543 [i]