Best Known (102−23, 102, s)-Nets in Base 7
(102−23, 102, 1530)-Net over F7 — Constructive and digital
Digital (79, 102, 1530)-net over F7, using
- net defined by OOA [i] based on linear OOA(7102, 1530, F7, 23, 23) (dual of [(1530, 23), 35088, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(7102, 16831, F7, 23) (dual of [16831, 16729, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7102, 16833, F7, 23) (dual of [16833, 16731, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(796, 16807, F7, 23) (dual of [16807, 16711, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(76, 26, F7, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(7102, 16833, F7, 23) (dual of [16833, 16731, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(7102, 16831, F7, 23) (dual of [16831, 16729, 24]-code), using
(102−23, 102, 16770)-Net over F7 — Digital
Digital (79, 102, 16770)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7102, 16770, F7, 23) (dual of [16770, 16668, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7102, 16819, F7, 23) (dual of [16819, 16717, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(7101, 16808, F7, 23) (dual of [16808, 16707, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(791, 16808, F7, 21) (dual of [16808, 16717, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7102, 16819, F7, 23) (dual of [16819, 16717, 24]-code), using
(102−23, 102, large)-Net in Base 7 — Upper bound on s
There is no (79, 102, large)-net in base 7, because
- 21 times m-reduction [i] would yield (79, 81, large)-net in base 7, but