Best Known (35−15, 35, s)-Nets in Base 128
(35−15, 35, 2343)-Net over F128 — Constructive and digital
Digital (20, 35, 2343)-net over F128, using
- 1281 times duplication [i] based on digital (19, 34, 2343)-net over F128, using
- net defined by OOA [i] based on linear OOA(12834, 2343, F128, 15, 15) (dual of [(2343, 15), 35111, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12834, 16402, F128, 15) (dual of [16402, 16368, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(12834, 16402, F128, 15) (dual of [16402, 16368, 16]-code), using
- net defined by OOA [i] based on linear OOA(12834, 2343, F128, 15, 15) (dual of [(2343, 15), 35111, 16]-NRT-code), using
(35−15, 35, 9363)-Net in Base 128 — Constructive
(20, 35, 9363)-net in base 128, using
- net defined by OOA [i] based on OOA(12835, 9363, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12835, 65542, S128, 15), using
- discarding parts of the base [i] based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding parts of the base [i] based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on OA(12835, 65542, S128, 15), using
(35−15, 35, 14474)-Net over F128 — Digital
Digital (20, 35, 14474)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12835, 14474, F128, 15) (dual of [14474, 14439, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12835, 16392, F128, 15) (dual of [16392, 16357, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([1,7]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12828, 16385, F128, 8) (dual of [16385, 16357, 9]-code), using the narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [1,7], and minimum distance d ≥ |{−7,−5,−3,…,7}|+1 = 9 (BCH-bound) [i]
- linear OA(1286, 7, F128, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,128)), using
- dual of repetition code with length 7 [i]
- construction X applied to C([0,7]) ⊂ C([1,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12835, 16392, F128, 15) (dual of [16392, 16357, 16]-code), using
(35−15, 35, large)-Net in Base 128 — Upper bound on s
There is no (20, 35, large)-net in base 128, because
- 13 times m-reduction [i] would yield (20, 22, large)-net in base 128, but