Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform

We enumerate the inequivalent self-dual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The On-Line Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a well-known interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we define PAR_IHN, peak-to-average power ratio with respect to the {I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South Korea, October 2004. 17 pages, 10 figure

Effects of interleukin-1β Inhibition on blood pressure, incident hypertension, and residual inflammatory risk

While hypertension and inflammation are physiologically inter-related, the effect of therapies that specifically target inflammation on blood pressure is uncertain. The recent CANTOS (Canakinumab Anti-inflammatory Thrombosis Outcomes Study) afforded the opportunity to test whether IL (interleukin)-1β inhibition would reduce blood pressure, prevent incident hypertension, and modify relationships between hypertension and cardiovascular events. CANTOS randomized 10 061 patients with prior myocardial infarction and hsCRP (high sensitivity C-reactive protein) ≥2 mg/L to canakinumab 50 mg, 150 mg, 300 mg, or placebo. A total of 9549 trial participants had blood pressure recordings during follow-up; of these, 80% had a preexisting diagnosis of hypertension. In patients without baseline hypertension, rates of incident hypertension were 23.4, 26.6, and 28.1 per 100-person years for the lowest to highest baseline tertiles of hsCRP (P>0.2). In all participants random allocation to canakinumab did not reduce blood pressure (P>0.2) or incident hypertension during the follow-up period (hazard ratio, 0.96 [0.85–1.08], P>0.2). IL-1β inhibition with canakinumab reduces major adverse cardiovascular event rates. These analyses suggest that the mechanisms underlying this benefit are not related to changes in blood pressure or incident hypertension

On Invariant Notions of Segre Varieties in Binary Projective Spaces

Invariant notions of a class of Segre varieties \Segrem(2) of PG(2^m - 1, 2) that are direct products of $m$ copies of PG(1, 2), $m$ being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains \Segrem(2) and is invariant under its projective stabiliser group \Stab{m}{2}. By embedding PG(2^m - 1, 2) into \PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant under \Stab{m}{2} as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as $m$ is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a \Stab{m}{2}-invariant geometric spread of lines of PG(2^m - 1, 2). This spread is also related with a \Stab{m}{2}-invariant non-singular Hermitian variety. The case $m=3$ is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under \Stab{3}{2}, while the points of PG(7, 2) form five orbits.Comment: 18 pages, 1 figure; v2 - version accepted in Designs, Codes and Cryptograph

Symplectic spreads and permutation polynomials

Every symplectic spread of PG(3, q), or equivalently every ovoid of Q(4, q), is shown to give a certain family of permutation polynomials of GF(q) and conversely. This leads to an algebraic proof of the existence of the Tits-Lüneburg spread of W(2 2h+1) and the Ree-Tits spread of W(3 2h+1), as well as to a new family of low-degree permutation polynomials over GF(3 2h+1)