On the number of binary signed digit representations of a given weight

Tůma, Jiří; Vábek, Jiří

doi:10.14712/1213-7243.2015.129

Cited by 6 publications

(6 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The result given here is exact, so improves upon the upper bound given in [18] by Tůma and Vábek for the number of i-bit BSD representations of an integer n of a given weight.…”

Section: The Weight Distribution Of the Bsd Representations Of An Int...supporting

confidence: 76%

“…2, the y axes represent the number of representations of a given integer n ∈ I k .The tick marks on the y-axes are the Fibonacci numbers, which are the maxima within a NAF-interval. The maxima have value F k 2 +1 and are reached in A k , as discussed in [5,6,13,18]. Relative maxima can also be seen within subintervals, and these are also Fibonacci numbers.…”

Section: Optimal Bsd Representations Of Integers In I Kmentioning

confidence: 85%

See 1 more Smart Citation

Binary Signed-Digit Integers and the Stern Polynomial

Monroe¹

2021

Preprint

View full text Add to dashboard Cite

The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in some depth. In this paper, we show previously unknown connections between BSD representations and the Stern polynomial.We derive a weight-distribution theorem for i-bit BSD representations of an integer n in terms of the coefficients and degrees of the terms of the Stern polynomial of 2 i − n.We then show new recursions on Stern polynomials, and from these and the weightdistribution theorem obtain similar BSD recursions and a fast O(n) algorithm that calculates the number and number of 0s of the optimal BSD representations of all of the integers of NAF-bitlength log(n) at once, which then may be compared.

show abstract

“…The result given here is exact, so improves upon the upper bound given in [18] by Tůma and Vábek for the number of i-bit BSD representations of an integer n of a given weight.…”

Section: The Weight Distribution Of the Bsd Representations Of An Int...supporting

confidence: 76%

Section: Optimal Bsd Representations Of Integers In I Kmentioning

confidence: 85%

Binary Signed-Digit Integers and the Stern Polynomial

Monroe¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…DEFINITION 2.1 (After Tůma and Vábek [13]). A binary signed-digit representation of an integer z is a string…”

Section: Normalised Additive Factorisationsupporting

confidence: 54%

Binary Signed-Digit Representations in Paperfolding

Bates

Bunder²,

Tognetti³

2022

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Wu et al showed this result, with the formula as in Theorem 48 for odd bitlength in [34], but did not exhaustively characterize such integers. In [31], Tůma et al also show this result, including the uniqueness. They approach the problem using transducers, as do Grabner et al [14], rather than the Stern polynomial approach we use.…”

Section: )) By Definition Of Vmentioning

confidence: 73%

Binary Signed-Digit Integers, the Stern Diatomic Sequence and Stern Polynomials

Monroe

2021

Preprint

View full text Add to dashboard Cite

Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit (BSD) representation of integers is used widely in efficient computation, coding theory and other applications. We link these two objects, showing that the number of i-bit binary signed-digit representations of an integer n < 2 i is the (2 i − n) th element in Stern's diatomic sequence. This correspondence makes the vast range of results known about the Stern diatomic sequence available for consideration in the study of binary signed-digit integers, and vice versa.Applications of this relationship discussed in this paper include a weight-distribution theorem for BSD representations, linking these representations to Stern polynomials, a recursion for the number of optimal BSD representations of an integer along with their Hamming weight, stemming from an easy recursion for the leading coefficients and degrees of Stern polynomials, and the identification of all integers having a maximal number of such representations.

show abstract

On the number of binary signed digit representations of a given weight

Cited by 6 publications

References 16 publications

Binary Signed-Digit Integers and the Stern Polynomial

Binary Signed-Digit Integers and the Stern Polynomial

Binary Signed-Digit Representations in Paperfolding

Binary Signed-Digit Integers, the Stern Diatomic Sequence and Stern Polynomials

Contact Info

Product

Resources

About