No abstract

Let A(n, d) (respectively A(n, d, w)) be the maximum possible number of codewords in a binary code (respectively binary constant-weight w code) of length n and minimum Hamming distance at least d.

a b s t r a c tLet R be a commutative ring with identity. We show that the Krull dimension of the power series ring R[[X ]] can be uncountably infinite, i.e., there exists an uncountably infinite chain of prime ideals in R[[X ]], even if dim R is finite. In fact, we show that dim R[[X ]] is uncountably infinite if R is a non-SFT ring, which is an improvement of Arnold's result.

Let α be a (finite or infinite) cardinal number. An ideal of a ring R is called an α-generated ideal if it can be generated by a set with cardinality at most α. A ring R is called an α-generated ring if every ideal of R is an α-generated ideal. When α is finite, the class of α-generated rings has been studied in literature by scholars such as I. S. Cohen and R. Gilmer. In this paper, the class of α-generated rings when α is infinite (in particular, when α = ℵ 0 , the smallest infinite cardinal number) is considered. Surprisingly, it is proved that the concepts "ℵ 0 -generated ring" and "Noetherian ring" are the same for the power series ring R[[X]]. In other words, if every ideal of R[[X]] is countably generated, then each of them is in fact finitely generated. This shows a strange behavior of the power series ring R[[X]] compared to that of the polynomial ring R[X]. Indeed, for any infinite cardinal number α, it is proved that R is an α-generated ring if and only if R[X] is an α-generated ring, which is an analogue of the Hilbert basis theorem stating that R is a Noetherian ring if and only if R[X] is a Noetherian ring. Let ᏻ be the ring of algebraic integers. Under the continuum hypothesis, we show that ᏻ [[X]] contains an |ᏻ[[X]]|-generated (and hence uncountably generated) ideal which is not a β-generated ideal for any cardinal number β < |ᏻ [[X]]| and that the concepts "ℵ 1 -generated ring" and "ℵ 0 -generated ring" are different for the power series ring R [[X]].

The Krongno Volcanic Geopark in Dak Nong province with a unique and largest volcanic cave system in SE Asia discovered in 2007 occupies an area about 2,000 km 2 in the Krong No, Cu Jut and Dak Mil districts and several neighboring communes in the Dak Song and Dak Glong districts. In 2016, Vietnamese archaeologists discovered a number of archaeological sites in hilly, farmland and riverine areas in the Krong-No Volcanic Geopark; however, none had been found in the volcanic caves. During the late December 2016-early January 2017 field trip, a survey team led by Vietnam National Museum of Nature, VAST, discovered a series of archaeological sites and relics with a relatively high density in volcanic caves in the Krongno Volcanic Geopark. The discovered archaeological remnants include (1) stone materials and tools such as dish-shaped tools, short axes, blade-ground short axes, oval axes and blade-ground oval axes; flake tools, stone flakes, stone slabs; anvils, graters, pestles, etc., hand-fitting sharp quartzite stones and pieces of loess; (2) pottery, containing various ceramic tools with different thicknesses, mostly fired at low temperature, crumbly, made of fine sandy loam, hand-kneaded, simple forms, mainly pots and containers having relatively sharp sculptings of varied patterns on the pottery shards such as dot-dash, dotted line, dashed line, twisted rope, etc.; (3) animal and teeth bones, including fragments of animal bone, or bones of prehistoric human (?). The bones are mostly decayed and fragile when dry; the hollow portions of bones are filled with porous clay slurry. In addition to the bone fragments, there are cheek teeth of animals some are fossilized. Preliminary determination suggested the teeth are of herbivores. Most of the archaeological artifacts were found in caves with flat floors, having mouths facing east, southeast or south heading to relatively wide areas to absorb the light, close to water sources and convenient to commute. These are the first prehistoric archaeological findings in volcanic caves in Vietnam, which will be studied in more detail by Vietnam National Museum of Nature, VAST, and other authorized agencies to highlight a new type of settlement, a new adaptive trend of prehistoric dwellers in the Central Highlands and to open a new archaeological study in the volcanic caves in Vietnam and Southeast Asia.