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Three parameters of Boolean functions related to their constancy on affine spaces
Advances in Mathematics of Communications ( IF 0.7 ) Pub Date : 2019-11-20 , DOI: 10.3934/amc.2020036
Claude Carlet , , Serge Feukoua ,

The $ k $-normality of Boolean functions is an important notion initially introduced by Dobbertin and studied in several papers. The parameter related to this notion is the maximal dimension of those affine spaces contained in the support $ supp(f) $ of the function or in its co-support $ cosupp(f) $. We denote it by $ norm\,(f) $ and call it the norm of $ f $.The norm concerns only the affine spaces contained in either the support or the co-support; the information it provides on $ f $ is then somewhat incomplete (for instance, two functions constant on a hyperplane will have the same very large parameter value, while they can have very different complexities). A second parameter which completes the information given by the first one is the minimum between the maximal dimension of those affine spaces contained in $ supp(f) $ and the maximal dimension of those contained in $ cosupp(f) $ (while $ norm\,(f) $ equals the maximum between these two maximal dimensions). We denote it by $ cons\,(f) $ and call it the (affine) constancy of $ f $.The value of $ cons\,(f) $ gives global information on $ f $, but no information on what happens around each point of $ supp(f) $ or $ cosupp(f) $. We define then its local version, equal to the minimum, when $ a $ ranges over $ \Bbb{F}_2^n $, of the maximal dimension of those affine spaces which contain $ a $ and on which $ f $ is constant. We denote it by $ stab\,(f) $ and call it the stability of $ f $.We study the properties of these three parameters. We have $ norm\,(f)\geq cons\,(f)\geq stab\,(f) $, then for determining to which extent these three parameters are distinct, we exhibit four infinite classes of Boolean functions, which show that all cases can occur, where each of these two inequalities can be strict or large.We consider the minimal value of $ stab\, (f) $ (resp. $ cons\,(f) $, $ norm\,(f) $), when $ f $ ranges over the Reed-Muller code $ RM(r,n) $ of length $ 2^n $ and order $ r $, and we denote it by $ stab\, _{RM(r,n)} $ (resp. $ cons\, _{RM(r,n)} $, $ norm\, _{RM(r,n)} $). We give upper bounds for each of these three integer sequences, and determine the exact values of $ stab\, _{RM(r,n)} $ and $ cons\, _{RM(r,n)} $ for $ r\in\{1,2,n-2,n-1,n\} $, and of $ norm\, _{RM(r,n)} $ for $ r = 1,2 $.

中文翻译:

布尔函数在仿射空间上的恒定性的三个参数

布尔函数的$ k-正态性是Dobbertin最初提出并在几篇论文中进行研究的重要概念。与该概念相关的参数是该函数的支撑$ supp(f)$或其协支撑$ cosupp(f)$中包含的那些仿射空间的最大尺寸。我们用$ norm \,(f)$表示它,并称其为$ f $的范数。该范数仅涉及支撑物或共同支撑物中包含的仿射空间;那么它在$ f $上提供的信息就有些不完整(例如,一个超平面上恒定的两个函数将具有相同的非常大的参数值,而它们的复杂度可能会非常不同)。完成由第一个参数给出的信息的第二个参数是$ supp(f)$中包含的仿射空间的最大维数与$ cosupp(f)$中包含的仿射空间的最大维数之间的最小值(而$ norm \ ,(f)$等于这两个最大尺寸之间的最大值)。我们用$ cons \,(f)$表示它,并将其称为$ f $的(仿射)不变性。$ cons \,(f)$的值提供了有关$ f $的全局信息,但没有有关发生的情况的信息在$ supp(f)$或$ cosupp(f)$的每个点周围。然后,我们定义其局部版本,即当$ a $的范围超过$ \ Bbb {F} _2 ^ n $时,包含$ a $且$ f $恒定的仿射空间的最大值。我们用$ stab \,(f)$表示它,并称其为$ f $的稳定性。我们研究了这三个参数的性质。我们有$ norm \,(f)\ geq cons \,(f)\ geq stab \,(f)$,然后为了确定这三个参数在不同程度上,我们展示了四个无限类的布尔函数,它们表明所有情况都可能发生,这两个不等式中的每个不等式都可以是严格的,也可以是大的。我们考虑$ stab \,(f)$(res $。 f $的范围为长度为2 ^ n $的Reed-Muller码$ RM(r,n)$并订购$ r $,我们用$ stab \,_ {RM(r,n)} $(分别为$ cons \,_ {RM(r,n)} $,$ norm \,_ {RM(r,n)} $)。我们为这三个整数序列的每一个给出上限,并为$ r确定$ stab \,_ {RM(r,n)} $和$ cons,_ {RM(r,n)} $的确切值\ in \ {1,2,n-2,n-1,n \} $和$ norm \的_ {RM(r,n)} $ for $ r = 1,2 $。然后,为了确定这三个参数在何种程度上不同,我们展示了四个无限类的布尔函数,它们表明所有情况都可能发生,这两个不等式都可以是严格的或大的。我们考虑$ stab \的最小值,(f)$(res $。cons \,(f)$,$ norm \,(f)$),当$ f $的范围为长度为$ 2的里德穆勒码$ RM(r,n)$ ^ n $并订购$ r $,我们用$ stab \,_ {RM(r,n)} $(分别为$ cons \,_ {RM(r,n)} $,$ norm \, _ {RM(r,n)} $)。我们为这三个整数序列的每一个给出上限,并为$ r确定$ stab \,_ {RM(r,n)} $和$ cons,_ {RM(r,n)} $的确切值\ in \ {1,2,n-2,n-1,n \} $和$ norm \的_ {RM(r,n)} $ for $ r = 1,2 $。然后,为了确定这三个参数在何种程度上不同,我们展示了四个无限类的布尔函数,它们表明所有情况都可能发生,这两个不等式都可以是严格的或大的。我们考虑$ stab \的最小值,(f)$(res $。cons \,(f)$,$ norm \,(f)$),当$ f $的范围为长度为$ 2的里德穆勒码$ RM(r,n)$ ^ n $并订购$ r $,我们用$ stab \,_ {RM(r,n)} $(分别为$ cons \,_ {RM(r,n)} $,$ norm \, _ {RM(r,n)} $)。我们为这三个整数序列的每一个给出上限,并为$ r确定$ stab \,_ {RM(r,n)} $和$ cons,_ {RM(r,n)} $的确切值\ in \ {1,2,n-2,n-1,n \} $和$ norm \的_ {RM(r,n)} $ for $ r = 1,2 $。这两个不等式中的每个不等式都可以是严格的,也可以是大的。我们考虑$ stab \,(f)$(res $。 f $的范围为长度为2 ^ n $的Reed-Muller码$ RM(r,n)$并订购$ r $,我们用$ stab \,_ {RM(r,n)} $(分别为$ cons \,_ {RM(r,n)} $,$ norm \,_ {RM(r,n)} $)。我们为这三个整数序列的每一个给出上限,并为$ r确定$ stab \,_ {RM(r,n)} $和$ cons,_ {RM(r,n)} $的确切值\ in \ {1,2,n-2,n-1,n \} $和$ norm \的_ {RM(r,n)} $为$ r = 1,2 $。这两个不等式中的每个不等式都可以是严格的,也可以是大的。我们考虑$ stab \,(f)$(res $。 f $的范围为长度为2 ^ n $的Reed-Muller码$ RM(r,n)$并订购$ r $,我们用$ stab \,_ {RM(r,n)} $(分别为$ cons \,_ {RM(r,n)} $,$ norm \,_ {RM(r,n)} $)。我们为这三个整数序列的每一个给出上限,并为$ r确定$ stab \,_ {RM(r,n)} $和$ cons,_ {RM(r,n)} $的确切值\ in \ {1,2,n-2,n-1,n \} $和$ norm \的_ {RM(r,n)} $ for $ r = 1,2 $。
更新日期:2019-11-20
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