Essential idempotents and simplex codes

We define essential idempotents in group algebras and use them to prove that every mininmal abelian non-cyclic code is a repetition code. Also we use them to prove that every minimal abelian code is equivalent to a minimal cyclic code of the same length. Finally, we show that a binary cyclic code is simplex if and only if is of length of the form $n=2^k-1$ and is generated by an essential idempotent

Pediatric patient with systemic lupus erythematosus & congenital acquired immunodeficiency syndrome: An unusual case and a review of the literature

The coexistence of systemic lupus erythematosus (SLE) in patients with congenital human immunodeficiency virus (HIV) infection is rare. This is a case report of a child diagnosed with SLE at nine years of age. She initially did well on non-steroidal anti-inflammatory agents, hydroxychloroquine, and steroids. She then discontinued her anti-lupus medications and was lost to follow-up. At 13 years of age, her lupus symptoms had resolved and she presented with intermittent fevers, cachexia, myalgias, arthralgias, and respiratory symptoms. Through subsequent investigations, the patient was ultimately diagnosed with congenitally acquired immunodeficiency syndrome (AIDS)

Early- Onset Stroke and Vasculopathy Associated with Mutations in ADA2

Adenosine deaminase 2 (ADA2) is an enzyme involved in purine metabolism and a growth factor that influences the development of endothelial cells and leukocytes. This study shows that defects in ADA2 cause recurrent fevers, vascular pathologic features, and mild immunodeficiency. Patients with autoinflammatory disease sometimes present with clinical findings that encompass multiple organ systems.(1) Three unrelated children presented to the National Institutes of Health (NIH) Clinical Center with intermittent fevers, recurrent lacunar strokes, elevated levels of acute-phase reactants, livedoid rash, hepatosplenomegaly, and hypogammaglobulinemia. Collectively, these findings do not easily fit with any of the known inherited autoinflammatory diseases. Hereditary or acquired vascular disorders can have protean manifestations yet be caused by mutations in a single gene. Diseases such as the Aicardi-Goutieres syndrome,(2),(3) polypoidal choroidal vasculopathy,(4) sickle cell anemia,(5) livedoid vasculopathy,(6) and the small-vessel vasculitides(7),(8) are examples of systemic ...</p