Best Known (45, 68, s)-Nets in Base 128
(45, 68, 190650)-Net over F128 — Constructive and digital
Digital (45, 68, 190650)-net over F128, using
- 1281 times duplication [i] based on digital (44, 67, 190650)-net over F128, using
- net defined by OOA [i] based on linear OOA(12867, 190650, F128, 23, 23) (dual of [(190650, 23), 4384883, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12867, 2097151, F128, 23) (dual of [2097151, 2097084, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12867, 2097151, F128, 23) (dual of [2097151, 2097084, 24]-code), using
- net defined by OOA [i] based on linear OOA(12867, 190650, F128, 23, 23) (dual of [(190650, 23), 4384883, 24]-NRT-code), using
(45, 68, 699053)-Net over F128 — Digital
Digital (45, 68, 699053)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12868, 699053, F128, 3, 23) (dual of [(699053, 3), 2097091, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12868, 2097159, F128, 23) (dual of [2097159, 2097091, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12868, 2097160, F128, 23) (dual of [2097160, 2097092, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12868, 2097160, F128, 23) (dual of [2097160, 2097092, 24]-code), using
- OOA 3-folding [i] based on linear OA(12868, 2097159, F128, 23) (dual of [2097159, 2097091, 24]-code), using
(45, 68, large)-Net in Base 128 — Upper bound on s
There is no (45, 68, large)-net in base 128, because
- 21 times m-reduction [i] would yield (45, 47, large)-net in base 128, but