Best Known (62, 62+21, s)-Nets in Base 27
(62, 62+21, 53145)-Net over F27 — Constructive and digital
Digital (62, 83, 53145)-net over F27, using
- 271 times duplication [i] based on digital (61, 82, 53145)-net over F27, using
- net defined by OOA [i] based on linear OOA(2782, 53145, F27, 21, 21) (dual of [(53145, 21), 1115963, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2782, 531451, F27, 21) (dual of [531451, 531369, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(2781, 531442, F27, 21) (dual of [531442, 531361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2782, 531451, F27, 21) (dual of [531451, 531369, 22]-code), using
- net defined by OOA [i] based on linear OOA(2782, 53145, F27, 21, 21) (dual of [(53145, 21), 1115963, 22]-NRT-code), using
(62, 62+21, 458879)-Net over F27 — Digital
Digital (62, 83, 458879)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2783, 458879, F27, 21) (dual of [458879, 458796, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2783, 531455, F27, 21) (dual of [531455, 531372, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(2781, 531441, F27, 21) (dual of [531441, 531360, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2783, 531455, F27, 21) (dual of [531455, 531372, 22]-code), using
(62, 62+21, large)-Net in Base 27 — Upper bound on s
There is no (62, 83, large)-net in base 27, because
- 19 times m-reduction [i] would yield (62, 64, large)-net in base 27, but