Best Known (34−15, 34, s)-Nets in Base 128
(34−15, 34, 2343)-Net over F128 — Constructive and digital
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
(34−15, 34, 9362)-Net in Base 128 — Constructive
(19, 34, 9362)-net in base 128, using
- net defined by OOA [i] based on OOA(12834, 9362, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12834, 65535, S128, 15), using
- discarding factors based on OA(12834, 65538, S128, 15), using
- discarding parts of the base [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- discarding factors based on OA(12834, 65538, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12834, 65535, S128, 15), using
(34−15, 34, 9964)-Net over F128 — Digital
Digital (19, 34, 9964)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12834, 9964, F128, 15) (dual of [9964, 9930, 16]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(12834, 16402, F128, 15) (dual of [16402, 16368, 16]-code), using
(34−15, 34, large)-Net in Base 128 — Upper bound on s
There is no (19, 34, large)-net in base 128, because
- 13 times m-reduction [i] would yield (19, 21, large)-net in base 128, but