Best Known (77−36, 77, s)-Nets in Base 25
(77−36, 77, 288)-Net over F25 — Constructive and digital
Digital (41, 77, 288)-net over F25, using
- net from sequence [i] based on digital (41, 287)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 41 and N(F) ≥ 288, using
(77−36, 77, 718)-Net over F25 — Digital
Digital (41, 77, 718)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2577, 718, F25, 36) (dual of [718, 641, 37]-code), using
- 81 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 11 times 0, 1, 22 times 0, 1, 38 times 0) [i] based on linear OA(2568, 628, F25, 36) (dual of [628, 560, 37]-code), using
- construction XX applied to C1 = C([623,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([623,34]) [i] based on
- linear OA(2566, 624, F25, 35) (dual of [624, 558, 36]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2566, 624, F25, 35) (dual of [624, 558, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2568, 624, F25, 36) (dual of [624, 556, 37]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2564, 624, F25, 34) (dual of [624, 560, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [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([623,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([623,34]) [i] based on
- 81 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 11 times 0, 1, 22 times 0, 1, 38 times 0) [i] based on linear OA(2568, 628, F25, 36) (dual of [628, 560, 37]-code), using
(77−36, 77, 300595)-Net in Base 25 — Upper bound on s
There is no (41, 77, 300596)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 437924 688611 053988 696338 732243 022636 605308 512803 977052 969257 582113 950004 254434 881161 059228 533905 825255 424065 > 2577 [i]