MinT - Best Known (131−14, 131, s)-Nets in Base 8

Best Known (131−14, 131, s)-Nets in Base 8

(131−14, 131, 2398108)-Net over F₈ — Constructive and digital

Digital (117, 131, 2398108)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (18, 25, 1366)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁵, 1366, F₈, 7, 7) (dual of [(1366, 7), 9537, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(8²⁵, 4099, F₈, 7) (dual of [4099, 4074, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(8²⁵, 4100, F₈, 7) (dual of [4100, 4075, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(8²⁵, 4096, F₈, 7) (dual of [4096, 4071, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(8²¹, 4096, F₈, 6) (dual of [4096, 4075, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(131−14, 131, large)-Net over F₈ — Digital

Digital (117, 131, large)-net over F₈, using

t-expansion [i] based on digital (113, 131, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (113, 132, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³², large, F₈, 19) (dual of [large, large−132, 20]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(8¹²⁹, large, F₈, 19) (dual of [large, large−129, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(131−14, 131, large)-Net in Base 8 — Upper bound on s

There is no (117, 131, large)-net in base 8, because

12 times m-reduction [i] would yield (117, 119, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (131−14, 131, s)-Nets in Base 8

(131−14, 131, 2398108)-Net over F8 — Constructive and digital

(131−14, 131, large)-Net over F8 — Digital

(131−14, 131, large)-Net in Base 8 — Upper bound on s

(131−14, 131, 2398108)-Net over F₈ — Constructive and digital

(131−14, 131, large)-Net over F₈ — Digital