Best Known (68−36, 68, s)-Nets in Base 25
(68−36, 68, 204)-Net over F25 — Constructive and digital
Digital (32, 68, 204)-net over F25, using
- t-expansion [i] based on digital (30, 68, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
(68−36, 68, 314)-Net over F25 — Digital
Digital (32, 68, 314)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2568, 314, F25, 2, 36) (dual of [(314, 2), 560, 37]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(2568, 628, F25, 36) (dual of [628, 560, 37]-code), using
(68−36, 68, 60111)-Net in Base 25 — Upper bound on s
There is no (32, 68, 60112)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 114806 192825 995188 993102 956366 602353 270294 391334 392852 382693 668887 739212 623139 355634 634266 605825 > 2568 [i]