# The Short Circuit

Mike Morgan's technical journal

### Selected Mapping with k-hot extension factor encoding of SI index

K-hot encoding of selected mapping index in orthogonal frequency division multiplexing for peak to average power reduction is a method of encoding the selected mapping index without the complexity of a reliable side channel, and was first proposed in [1].   In [2], I review and expand on this method .

In SLM-KHE, SI indexes are encoded by creating energy disparity in sub-vectors, before mapping to several OFDM symbols.  A selector transmits an OFDM symbol with the minimal peak to average power ratio.  At the receiver, the OFDM symbol is demodulated by DFT, energy disparity is measured and used to decode the SI index used by the transmitter, and original vector phases are restored based on the SI index detected.  A block diagram from [2] is shown below.

The full paper in pdf format can be found in [2] below, and incudes brief backgrounds on orthogonal frequency division multiplexing (OFDM), the peak to average power problem in OFDM, selected mapping with side information (SLM), selected mapping with k-hot encoding (SLM-KHE), analysis descriptions, results, and other references.  Correspondence on this topic is welcome.

References

[1] S. Y. Le Goff, S. S. Al-Samahi, B. K. Khoo, C. C. Tsimenidis, and B. S. Sharif, "Selected Mapping without Side Information for PAPR Reduction in OFDM," Wireless Communications, IEEE Transactions on, vol. 8, no. 7, pp. 3320-3325, 2009.

[2] Morgan, Mike Review of Selected Mapping with k-hot extension factor encoding of SI index, self-published, April 8, 2017

### Speed solving the cube by Dan Harris, errata

My son noticed some errors in Dan Harris' book, "Speedsolving the Cube:  Easy-to-Follow, Step-by-Step Instructions for Many Popular 3-D puzzles."

The book has an errata, but the website where the errata was published is down and has been for some time.  So I am going to republish it here, for the benefit of others who may have the book and no list of errors.

### Closed form solution to Josephus problem

My 9 year old son Alexander came up with a couple of closed form solutions to the Joesphus problem.  Really.  If this helps you, please leave a comment, he will enjoy it.

A discussion of his solution (conjectures) and an attempted proof by me of one of them can be found as the first link in the references.

His main conjecture is:

$J(n)=2(n-2^{\lfloor\log_2 n \rfloor})+1, \mbox{ for } {n} \ge {1}$

where

$floor(x)=\lfloor{x}\rfloor$

is the largest integer not greater than x.