MinT - Best Known (102−10, 102, s)-Nets in Base 8

Best Known (102−10, 102, s)-Nets in Base 8

(102−10, 102, 3617595)-Net over F₈ — Constructive and digital

Digital (92, 102, 3617595)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (23, 28, 262155)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁸, 262155, F₈, 6, 5) (dual of [(262155, 6), 1572902, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(8²⁸, 262156, F₈, 2, 5) (dual of [(262156, 2), 524284, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(8², 9, F₈, 2, 2) (dual of [(9, 2), 16, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;16,8) [i]
        linear OOA(8²⁶, 262147, F₈, 2, 5) (dual of [(262147, 2), 524268, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(8²⁶, 524294, F₈, 5) (dual of [524294, 524268, 6]-code), using
        trace code [i] based on linear OA(64¹³, 262147, F₆₄, 5) (dual of [262147, 262134, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 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 (64, 74, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(102−10, 102, 3879730)-Net in Base 8 — Constructive

(92, 102, 3879730)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (23, 28, 524290)-net in base 8, using
  - base change [i] based on digital (16, 21, 524290)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16²¹, 524290, F₁₆, 5, 5) (dual of [(524290, 5), 2621429, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(16²¹, 1048581, F₁₆, 5) (dual of [1048581, 1048560, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(16²¹, 1048576, F₁₆, 5) (dual of [1048576, 1048555, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(16¹⁶, 1048576, F₁₆, 4) (dual of [1048576, 1048560, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 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 (64, 74, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(102−10, 102, large)-Net over F₈ — Digital

Digital (92, 102, large)-net over F₈, using

t-expansion [i] based on digital (90, 102, large)-net over F₈, using
- 3 times m-reduction [i] based on digital (90, 105, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁵, large, F₈, 15) (dual of [large, large−105, 16]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(102−10, 102, large)-Net in Base 8 — Upper bound on s

There is no (92, 102, large)-net in base 8, because

8 times m-reduction [i] would yield (92, 94, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (102−10, 102, s)-Nets in Base 8

(102−10, 102, 3617595)-Net over F8 — Constructive and digital

(102−10, 102, 3879730)-Net in Base 8 — Constructive

(102−10, 102, large)-Net over F8 — Digital

(102−10, 102, large)-Net in Base 8 — Upper bound on s

(102−10, 102, 3617595)-Net over F₈ — Constructive and digital

(102−10, 102, large)-Net over F₈ — Digital