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We delve into the connection between base $frac{3}{2}$ and the greedy partition of non-negative integers into 3-free sequences. Specifically, we find a fractal structure on strings written with digits 0, 1, and 2. We use this structure to prove that the even non-negative integers written in base $frac{3}{2}$ and then interpreted in base 3 form the Stanley cross-sequence, where the Stanley cross-sequence comprises the first terms of the infinitely many sequences that are formed by the greedy partition of non-negative integers into 3-free sequences.
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We discuss properties of integers in base 3/2. We also introduce many new sequences related to base 3/2. Some sequences discuss patterns related to integers in base 3/2. Other sequence are analogues of famous base-10 sequences: we discuss powers of 3 and 2, Look-and-say, and sorted and reverse sorted Fibonaccis. The eventual behavior of sorted and reverse sorted Fibs leads to special Pinocchio and Oihcconip sequences respectively.
We give a characterization of all matrices $A,B,C in mathbb{F}_{2}^{m times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,Cinmathbb{F}_{2}^{mathbb{N}timesmathbb{N}}$ which generate a $(0,2)$-sequence in base $2$.
We discuss two different systems of number representations that both can be called base 3/2. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a connection between these systems and 3-free sequences.
We determine the image and fibres for solvable base change.
In this study, we propose partitioned complementary sequences (CSs) where the gaps between the clusters encode information bits to achieve low peak-to-average-power ratio (PAPR) orthogonal frequency division multiplexing (OFDM) symbols. We show that the partitioning rule without losing the feature of being a CS coincides with the non-squashing partitions of a positive integer and leads to a symmetric separation of clusters. We analytically derive the number of partitioned CSs for given bandwidth and a minimum distance constraint and obtain the corresponding recursive methods for enumerating the values of separations. We show that partitioning can increase the spectral efficiency (SE) without changing the alphabet of the nonzero elements of the CS, i.e., standard CSs relying on Reed-Muller (RM) code. We also develop an encoder for partitioned CSs and a maximum-likelihood-based recursive decoder for additive white Gaussian noise (AWGN) and fading channels. Our results indicate that the partitioned CSs under a minimum distance constraint can perform similar to the standard CSs in terms of average block error rate (BLER) and provide a higher SE at the expense of a limited signal-to-noise ratio (SNR) loss.
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