Best Known (77−23, 77, s)-Nets in Base 27
(77−23, 77, 1792)-Net over F27 — Constructive and digital
Digital (54, 77, 1792)-net over F27, using
- 272 times duplication [i] based on digital (52, 75, 1792)-net over F27, using
- net defined by OOA [i] based on linear OOA(2775, 1792, F27, 23, 23) (dual of [(1792, 23), 41141, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2775, 19713, F27, 23) (dual of [19713, 19638, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2775, 19716, F27, 23) (dual of [19716, 19641, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- linear OA(2767, 19684, F27, 23) (dual of [19684, 19617, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2743, 19684, F27, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(278, 32, F27, 7) (dual of [32, 24, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 38, F27, 7) (dual of [38, 30, 8]-code), using
- extended algebraic-geometric code AGe(F,30P) [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- discarding factors / shortening the dual code based on linear OA(278, 38, F27, 7) (dual of [38, 30, 8]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2775, 19716, F27, 23) (dual of [19716, 19641, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2775, 19713, F27, 23) (dual of [19713, 19638, 24]-code), using
- net defined by OOA [i] based on linear OOA(2775, 1792, F27, 23, 23) (dual of [(1792, 23), 41141, 24]-NRT-code), using
(77−23, 77, 35628)-Net over F27 — Digital
Digital (54, 77, 35628)-net over F27, using
(77−23, 77, large)-Net in Base 27 — Upper bound on s
There is no (54, 77, large)-net in base 27, because
- 21 times m-reduction [i] would yield (54, 56, large)-net in base 27, but