Best Known (146, 173, s)-Nets in Base 8
(146, 173, 161323)-Net over F8 — Constructive and digital
Digital (146, 173, 161323)-net over F8, using
- net defined by OOA [i] based on linear OOA(8173, 161323, F8, 27, 27) (dual of [(161323, 27), 4355548, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8173, 2097200, F8, 27) (dual of [2097200, 2097027, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8173, 2097205, F8, 27) (dual of [2097205, 2097032, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(8162, 2097152, F8, 27) (dual of [2097152, 2096990, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(811, 53, F8, 6) (dual of [53, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(811, 63, F8, 6) (dual of [63, 52, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(811, 63, F8, 6) (dual of [63, 52, 7]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(8173, 2097205, F8, 27) (dual of [2097205, 2097032, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8173, 2097200, F8, 27) (dual of [2097200, 2097027, 28]-code), using
(146, 173, 2097206)-Net over F8 — Digital
Digital (146, 173, 2097206)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8173, 2097206, F8, 27) (dual of [2097206, 2097033, 28]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8172, 2097204, F8, 27) (dual of [2097204, 2097032, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(8162, 2097152, F8, 27) (dual of [2097152, 2096990, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(810, 52, F8, 6) (dual of [52, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(8172, 2097205, F8, 26) (dual of [2097205, 2097033, 27]-code), using Gilbert–Varšamov bound and bm = 8172 > Vbs−1(k−1) = 9500 815193 014620 997069 268746 727933 157413 684673 545155 958853 272828 168507 691871 221601 046024 092865 757511 128287 613836 009924 065728 266684 400524 396964 434189 400892 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8172, 2097204, F8, 27) (dual of [2097204, 2097032, 28]-code), using
- construction X with Varšamov bound [i] based on
(146, 173, large)-Net in Base 8 — Upper bound on s
There is no (146, 173, large)-net in base 8, because
- 25 times m-reduction [i] would yield (146, 148, large)-net in base 8, but