Best Known (44−19, 44, s)-Nets in Base 25
(44−19, 44, 200)-Net over F25 — Constructive and digital
Digital (25, 44, 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
(44−19, 44, 838)-Net over F25 — Digital
Digital (25, 44, 838)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2544, 838, F25, 19) (dual of [838, 794, 20]-code), using
- 203 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 57 times 0, 1, 109 times 0) [i] based on linear OA(2537, 628, F25, 19) (dual of [628, 591, 20]-code), using
- construction XX applied to C1 = C([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- 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(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2537, 624, F25, 19) (dual of [624, 587, 20]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [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(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([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- 203 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 57 times 0, 1, 109 times 0) [i] based on linear OA(2537, 628, F25, 19) (dual of [628, 591, 20]-code), using
(44−19, 44, 825248)-Net in Base 25 — Upper bound on s
There is no (25, 44, 825249)-net in base 25, because
- 1 times m-reduction [i] would yield (25, 43, 825249)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1 292480 277032 202250 686423 352864 631506 758353 427904 369568 237785 > 2543 [i]