John    Error correcting codes are most simply described in terms of the errors they can certainly correct. For example, the Hadamard code used for the...

John    Hadamard matrices are in the news following the recent announcement of a newly discovered Hadamard matrix. I’ve written three posts on Hadamard...

John    Yesterday Levent Alpöge announced that he and his colleagues had discovered a new Hadamard matrix using Claude AI. That motivated a post I wrote this...

John    NASA set Mariner 9 to photograph Mars in 1971. The images had to be encoded for transmission using an error-correcting code, otherwise they would be...

John    A Hadamard matrix is an orthogonal matrix whose entries are all either 1 or’ 1. For example is a Hadamard matrix of order 2. True to Stigler’s law of...

John    Suppose you’ve never worked at the command line and you’re reading a book about the bash shell. You read that is a shortcut to refer to the last word...

John    I was reading a blog post on boat names because it was on Hacker News this morning. The post contained a link to a data set on dog names in NYC and I...

John    Suppose you were able to go back in time, to an era before computers, and give someone contemporary cryptography. Encryption methods that are...

John    The following extraordinary paragraph comes from Knuth’s TAOCP Volume 4A, right before the last set of exercises. Many of the exercises below ask a...

John    In science and engineering classes, you might describe a system using Newton’s laws and end up with a differential equation. You then solve the...

John    I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions....

John    At the end of my post on how not to calculate cosine I said that the first step in calculating cosine, particularly cosine of a large number, would...

John    I have a program that shares files between my laptop and my phone. It works well, except for apostrophes. When I type an apostrophe ’ on my laptop,...

John    Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don...

John    In a footnote to the previous post, I said that Python’s math library can calculate the logarithm of extremely large numbers but not the cosine. This...

John    The previous post pointed out that the following code such as the following unexpectedly works. from math import log, factorial log(factorial(1000))...

John    Last week I wrote a post on hiding cryptographic keys in decks of cards. I wrote some code for that post that shouldn’t work, but before fixing I...

John    A few days ago I wrote a post on counting rooted trees. That post looked at the sequence c(n) which counts the number of rooted trees with n nodes....

John    After writing the previous post about metallic ratios, I thought about the analogy to alchemy and the attempt to make precious metals out of base...

John    The golden ratio is the first and best known of the metallic ratios. I’ve written about the silver ratio a few times, most recently here. And I’ve...

John    Yesterday I wrote that a lot of the special functions that pop up in mathematical physics are solutions to second order linear differential equations...

John    In this post I mentioned two series which I denoted t(n) and c(n). The former is the number of unlabeled rooted trees with n nodes. The latter is the...

John    Second order linear differential equations with polynomial coefficients form their own area of study. This seems like a narrow class of equations,...

John    Combinatorial problems can be interesting for their own sake, but they are more interesting when there is a connection to a problem outside...

John    The textbook version of the Runge-Kutta method for solving differential equations has 4 stages and has 4th order error. For lower order versions of...

John    I was digging into the Runge-Kutta method for solving differential equations and a line from [1] piqued my curiosity. These calculations, which are...

John    A couple years ago I wrote about how to compute the inverse of factorial. I used that code in writing the previous post because the post required...

John    The previous post looked at the idea of storing a cryptographic key in the order of a deck of cards. A deck of 52 cards can store 225 bits of data...

John    The latest issue of Paged Out has an article by Stephen Hewitt An off-line backup of your cryptographic key using playing cards. The idea is to use a...

John    My post from yesterday on permutation roots ends with a Mathematica code for finding the probability that a permutation of n elements has a kth root....