Best Known (8, 15, s)-Nets in Base 128
(8, 15, 5463)-Net over F128 — Constructive and digital
Digital (8, 15, 5463)-net over F128, using
- 1281 times duplication [i] based on digital (7, 14, 5463)-net over F128, using
- net defined by OOA [i] based on linear OOA(12814, 5463, F128, 7, 7) (dual of [(5463, 7), 38227, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12814, 16390, F128, 7) (dual of [16390, 16376, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1289, 16385, F128, 5) (dual of [16385, 16376, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(12814, 16390, F128, 7) (dual of [16390, 16376, 8]-code), using
- net defined by OOA [i] based on linear OOA(12814, 5463, F128, 7, 7) (dual of [(5463, 7), 38227, 8]-NRT-code), using
(8, 15, 16300)-Net over F128 — Digital
Digital (8, 15, 16300)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12815, 16300, F128, 7) (dual of [16300, 16285, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(12815, 16388, F128, 7) (dual of [16388, 16373, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12812, 16385, F128, 4) (dual of [16385, 16373, 5]-code), using the narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [1,3], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- linear OA(1282, 3, F128, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,128)), using
- dual of repetition code with length 3 [i]
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12815, 16388, F128, 7) (dual of [16388, 16373, 8]-code), using
(8, 15, 21845)-Net in Base 128 — Constructive
(8, 15, 21845)-net in base 128, using
- net defined by OOA [i] based on OOA(12815, 21845, S128, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12815, 65536, S128, 7), using
- discarding factors based on OA(12815, 65538, S128, 7), using
- discarding parts of the base [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- discarding factors based on OA(12815, 65538, S128, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12815, 65536, S128, 7), using
(8, 15, 32768)-Net in Base 128
(8, 15, 32768)-net in base 128, using
- net defined by OOA [i] based on OOA(12815, 32768, S128, 9, 7), using
- OOA stacking with additional row [i] based on OOA(12815, 32769, S128, 3, 7), using
- discarding parts of the base [i] based on linear OOA(25613, 32769, F256, 3, 7) (dual of [(32769, 3), 98294, 8]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25613, 32769, F256, 2, 7) (dual of [(32769, 2), 65525, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25613, 32769, F256, 2, 7) (dual of [(32769, 2), 65525, 8]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(25613, 32769, F256, 3, 7) (dual of [(32769, 3), 98294, 8]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12815, 32769, S128, 3, 7), using
(8, 15, large)-Net in Base 128 — Upper bound on s
There is no (8, 15, large)-net in base 128, because
- 5 times m-reduction [i] would yield (8, 10, large)-net in base 128, but