Best Known (62, 62+21, s)-Nets in Base 8
(62, 62+21, 819)-Net over F8 — Constructive and digital
Digital (62, 83, 819)-net over F8, using
- 81 times duplication [i] based on digital (61, 82, 819)-net over F8, using
- net defined by OOA [i] based on linear OOA(882, 819, F8, 21, 21) (dual of [(819, 21), 17117, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(882, 8191, F8, 21) (dual of [8191, 8109, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 8194, F8, 21) (dual of [8194, 8112, 22]-code), using
- trace code [i] based on linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 8194, F8, 21) (dual of [8194, 8112, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(882, 8191, F8, 21) (dual of [8191, 8109, 22]-code), using
- net defined by OOA [i] based on linear OOA(882, 819, F8, 21, 21) (dual of [(819, 21), 17117, 22]-NRT-code), using
(62, 62+21, 8198)-Net over F8 — Digital
Digital (62, 83, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(883, 8198, F8, 21) (dual of [8198, 8115, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(882, 8196, F8, 21) (dual of [8196, 8114, 22]-code), using
- trace code [i] based on linear OA(6441, 4098, F64, 21) (dual of [4098, 4057, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(6441, 4098, F64, 21) (dual of [4098, 4057, 22]-code), using
- linear OA(882, 8197, F8, 20) (dual of [8197, 8115, 21]-code), using Gilbert–Varšamov bound and bm = 882 > Vbs−1(k−1) = 20 954701 272470 196697 694744 755038 080263 856542 204191 735188 348862 794557 145600 [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(882, 8196, F8, 21) (dual of [8196, 8114, 22]-code), using
- construction X with Varšamov bound [i] based on
(62, 62+21, large)-Net in Base 8 — Upper bound on s
There is no (62, 83, large)-net in base 8, because
- 19 times m-reduction [i] would yield (62, 64, large)-net in base 8, but