Best Known (29, 29+10, s)-Nets in Base 27
(29, 29+10, 106291)-Net over F27 — Constructive and digital
Digital (29, 39, 106291)-net over F27, using
- net defined by OOA [i] based on linear OOA(2739, 106291, F27, 10, 10) (dual of [(106291, 10), 1062871, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2739, 531455, F27, 10) (dual of [531455, 531416, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(2737, 531441, F27, 10) (dual of [531441, 531404, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2725, 531441, F27, 7) (dual of [531441, 531416, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- OA 5-folding and stacking [i] based on linear OA(2739, 531455, F27, 10) (dual of [531455, 531416, 11]-code), using
(29, 29+10, 531455)-Net over F27 — Digital
Digital (29, 39, 531455)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2739, 531455, F27, 10) (dual of [531455, 531416, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(2737, 531441, F27, 10) (dual of [531441, 531404, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2725, 531441, F27, 7) (dual of [531441, 531416, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
(29, 29+10, large)-Net in Base 27 — Upper bound on s
There is no (29, 39, large)-net in base 27, because
- 8 times m-reduction [i] would yield (29, 31, large)-net in base 27, but