In the rank modulation scheme for flash memories, permutation codes have been studied. this paper, we study perfect in $$S_n$$ , set of all permutations on n elements, under Kendall $$\tau $$ -metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving nonexistence -metric, more values n. Specifically, present polynomial representation size a ball -metric some radius r, ...

Questions about order versus disorder in systems and models have been fascinating scientists over the years. In Computer Science, order is intimately related to sorting, commonly meant as the task of arranging keys in increasing or decreasing order with respect to an underlying total order relation. The sorted organization is amenable for searching a set of n keys, since each search requires Θ(...

In the fillable array problem one must maintain an array A[1..n] of w-bit entries subject to random access reads and writes, and also a fill(∆) operation which sets every entry of A to some ∆ ∈ {0, . . . , 2− 1}. We show that with just one bit of redundancy, i.e. a data structure using nw + 1 bits of memory, read/fill can be implemented in worst case constant time, and write can be implemented ...

We explore the concept of tonal signatures developed and put into musical practice by one of us (Mezzadri). A tonal signature of a scale S is a minimal subset of notes within S that is not contained in any scale S' different from S. We present a set covering model to find a smallest signature. We also show that the signatures of a scale are the prime implicants of a suitable monotone Boolean fu...

We propose a general framework for differential and linear cryptanalysis of block ciphers when the block is not a bitstring. We prove piling-up lemmas for the generalized differential probability and the linear potential, and we study their lower bounds and average value, in particular in the case of permutations of Fp. Using this framework, we describe a toy cipher, that operates on blocks of ...

Usually, binary search only makes sense in sorted arrays. We show that insertion sort based on repeated “binary searches” in an initially unsorted array also sorts n elements in time Θ(n log n). If n is a power of two then the expected termination point of a binary search in a random permutation of n elements is exactly the cell where the element should be if the array was sorted. We further sh...

We have developed a bit-reversal algorithm (BRAVO) using vector permute operations, which is optimal in the number of permutations, and its cache-optimal version (COBRAVO). Our implementation on PowerMac G5 shows 2– 4.5 fold improvement for small data sets and 15–75% improvement for large data sets (depending on the data element size) over the best known approach (COBRA).

Bit reversal, hierarchical memory, Fast Fourier Transform Many versions of the Fast Fourier Transform require a reordering of either the input or the output data that corresponds to reversing the order of the bits in the array index. There has been a surprisingly large number of papers on this subject in the recent literature. This paper collects 30 methods for bit reversing an array. Each meth...